Eureka Math, A Story of Functionsยฎ
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Eureka Mathโข Homework Helper
2015โ2016
Algebra IIModule 1
Lessons 1โ40
2015-16 M1 ALGEBRA II
Lesson 1: Successive Differences in Polynomials
I calculate first differences by subtracting the outputs for successive inputs. The second differences are the differences of successive first differences, and the third differences are differences between successive second differences.
Lesson 1: Successive Differences in Polynomials
Investigate Successive Differences in Sequences
Create a table to find the third differences for the polynomial 45๐ฅ๐ฅ โ 10๐ฅ๐ฅ2 + 3๐ฅ๐ฅ3 for integer values of ๐ฅ๐ฅ from 0 to 4.
๐ฅ๐ฅ ๐ฆ๐ฆ First Differences Second Differences
Third Differences
0 ๐๐
๐๐๐๐ โ ๐๐ = ๐๐๐๐
1 ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ = โ๐๐
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ ๐๐๐๐ โ (โ๐๐) = ๐๐๐๐
2 ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
3 ๐๐๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐
4 ๐๐๐๐๐๐
I notice the third differences are constant, which should be true for a polynomial of degree 3.
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1
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 1: Successive Differences in Polynomials
I found the second differences to be 10, and I know that the second differences are equal to 2๐๐, so it must be that ๐๐ = 5. I know the ๐ฆ๐ฆ-intercept is equal to ๐๐ and 5(0)2 + ๐๐(0) + ๐๐ = 6, so ๐๐ = 6.
I solved ๐ฆ๐ฆ = 5๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + 6 when ๐ฅ๐ฅ = 1 and ๐ฆ๐ฆ = 9 to find the value of ๐๐.
To model the ordered pairs with a quadratic equation, I need to find values of ๐๐, ๐๐, and ๐๐ in the equation ๐ฆ๐ฆ = ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐.
Write Explicit Polynomial Expressions for Sequences by Investigating Their Successive Differences
Show that the set of ordered pairs (๐ฅ๐ฅ,๐ฆ๐ฆ) in the table below satisfies a quadratic relationship. Find the equation of the form ๐ฆ๐ฆ = ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ that all of the ordered pairs satisfy.
๐ฅ๐ฅ 0 1 2 3 4 ๐ฆ๐ฆ 6 9 22 45 78
๐ฅ๐ฅ ๐ฆ๐ฆ First Differences Second Differences Third Differences 0 6
๐๐
1 9 ๐๐๐๐
๐๐๐๐
2 22 ๐๐๐๐
๐๐๐๐
3 45 ๐๐๐๐
๐๐๐๐
4 78
๐๐ = ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐ ๐๐ = ๐๐ and ๐๐ = ๐๐
๐๐ = ๐๐(๐๐)๐๐ + ๐๐(๐๐) + ๐๐ ๐๐ = ๐๐(๐๐)๐๐ + ๐๐ + ๐๐ ๐๐ = ๐๐๐๐ + ๐๐ ๐๐ = โ๐๐ ๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐
I know that if the second differences are constant, and the first differences are not, then the relationship is quadratic.
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2
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 2: The Multiplication of Polynomials
Now that I have combined like terms, I write the product as the sum of the terms from highest to lowest degree.
The top row contains the terms of the first polynomial.
The right-hand column contains the terms of the second polynomial.
I multiply the terms in the corresponding row and column to fill in each cell in the table.
Lesson 2: The Multiplication of Polynomials
Use a Table to Multiply Two Polynomials
1. Use the tabular method to multiply (5๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 3)(2๐ฅ๐ฅ3 โ 5๐ฅ๐ฅ2 + 4๐ฅ๐ฅ + 1), and combine like terms.
๐๐๐๐๐๐ ๐๐ ๐๐
๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐
โ๐๐๐๐๐๐๐๐ โ๐๐๐๐๐๐ โ๐๐๐๐๐๐๐๐ โ๐๐๐๐๐๐
๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐
๐๐๐๐๐๐ ๐๐ ๐๐ ๐๐
๏ฟฝ๐๐๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐
๐๐๐๐๐๐๐๐
โ๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐
๐๐ ๐๐๐๐๐๐ โ๐๐๐๐๐๐
For example, ๐ฅ๐ฅ โ 2๐ฅ๐ฅ3 = 2๐ฅ๐ฅ4.
Also, 3 โ (โ5๐ฅ๐ฅ2) = โ15๐ฅ๐ฅ2 .
I add the like terms found along the upper right to lower left diagonals in the table. For example, 6๐ฅ๐ฅ3 โ5๐ฅ๐ฅ3 + 20๐ฅ๐ฅ3 = 21๐ฅ๐ฅ3.
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3
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 2: The Multiplication of Polynomials
I recognize this pattern: (๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ๐๐ + ๐ฅ๐ฅ๐๐โ1 + ๐ฅ๐ฅ๐๐โ2 + โฏ+ ๐ฅ๐ฅ + 1) = ๐ฅ๐ฅ๐๐+1 โ 1
Use the Distributive Property to Multiply Two Polynomials
2. Use the distributive property to multiply (6๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 2)(7๐ฅ๐ฅ + 5), and then combine like terms.
๏ฟฝ๐๐๐๐๐๐ + ๐๐ โ ๐๐๏ฟฝ(๐๐๐๐ + ๐๐) = ๐๐๐๐๐๐(๐๐๐๐ + ๐๐) + ๐๐(๐๐๐๐ + ๐๐) โ ๐๐(๐๐๐๐ + ๐๐) = ๐๐๐๐๐๐(๐๐๐๐) + ๐๐๐๐๐๐(๐๐) + ๐๐(๐๐๐๐) + ๐๐(๐๐) โ ๐๐(๐๐๐๐) โ ๐๐(๐๐) = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ +๐๐๐๐๐๐+ ๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐
Apply Identities to Multiply Polynomials
Multiply the polynomials in each row of the table.
3. (๐ฅ๐ฅ + 2๐ฆ๐ฆ2)(๐ฅ๐ฅ โ 2๐ฆ๐ฆ2) (๐๐)๐๐ โ (๐๐๐๐๐๐)๐๐ = ๐๐๐๐ โ ๐๐๐๐๐๐
4. (3๐ฅ๐ฅ โ 7๐ฆ๐ฆ)2 (๐๐๐๐)๐๐ + ๐๐(๐๐๐๐)(โ๐๐๐๐) + (โ๐๐๐๐)๐๐
= ๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐
5. (๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ6 + ๐ฅ๐ฅ5 + ๐ฅ๐ฅ4 + ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 1) ๐๐๐๐+๐๐ โ ๐๐ = ๐๐๐๐ โ ๐๐
I distribute each term in the first polynomial to the second polynomial. Then I multiply each term of the first polynomial by each term in the second polynomial.
If I recognize an identity, I can use the formula associated with it to multiply polynomials. If I do not recognize an identity, I can multiply polynomials using the tabular method or distributive property.
I recognize this product as a difference of squares: (๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐2 โ ๐๐2. In this case, ๐๐ is ๐ฅ๐ฅ and ๐๐ is 2๐ฆ๐ฆ2.
I know that (๐๐ โ ๐๐)2 = ๐๐2 โ 2๐๐๐๐ + ๐๐2. In this case, ๐๐ is 3๐ฅ๐ฅ and ๐๐ is โ7๐ฆ๐ฆ.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 II
Lesson 3: The Division of Polynomials
I know the leading term in the dividend is equal to the diagonal sum and, therefore, belongs in this cell.
I need to find an expression that is equal to 2๐ฅ๐ฅ5 when it is multiplied by ๐ฅ๐ฅ3. I can then multiply this expression by the terms in the divisor to complete the first column of the table.
These are the terms of the divisor.
I need to write the terms of the dividend along the outside of the table aligned with the upper-right to lower-left diagonals, including placeholders. These terms represent the diagonal sums.
Lesson 3: The Division of Polynomials
Use a Table to Divide Two Polynomials
Use the reverse tabular method to find the quotient: (2๐ฅ๐ฅ5 + 7๐ฅ๐ฅ4 + 22๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ + 12) รท (๐ฅ๐ฅ3 + 4๐ฅ๐ฅ2 + 3).
Step 1
๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐
๐๐
Step 2
๐๐๐๐๐๐
๐๐๐๐๐๐ ๐๐๐๐
๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐
I need to use a placeholder here because the divisor does not contain a linear term.
๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐ โ๐๐๐๐ ๐๐๐๐๐๐๐๐
๐๐๐๐ โ๐๐๐๐ ๐๐๐๐๐๐๐๐
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Homework Helper A Story of FunctionsALGEBRA II
ALG II-M1-HWH-1.3.0-08.2015
๐๐๐๐
๐๐๐๐
๐๐
2015-16 M1 II
Lesson 3: The Division of Polynomials
Step 4
๐๐๐๐๐๐ โ๐๐ +๐๐
๐๐๐๐๐๐ โ๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐
๐๐๐๐๐๐ โ๐๐๐๐๐๐ ๐๐๐๐๐๐
๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐
๐๐๐๐๐๐ โ๐๐๐๐
๏ฟฝ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐๏ฟฝ รท ๏ฟฝ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๏ฟฝ = ๐๐๐๐๐๐ โ ๐๐ + ๐๐
Step 3
๐๐๐๐๐๐ โ๐๐
๐๐๐๐๐๐ โ๐๐๐๐๐๐ ๐๐๐๐
๐๐๐๐๐๐ ๐๐๐๐๐๐
๐๐๐๐๐๐ ๐๐๐๐
๐๐๐๐๐๐ ๐๐
๐๐๐๐ โ๐๐๐๐ ๐๐๐๐๐๐๐๐
I know this value, added to 8๐ฅ๐ฅ4, must equal the diagonal sum, 7๐ฅ๐ฅ4.
1๐ฅ๐ฅ 4
I need to find an expression that equals when multiplied by ๐ฅ๐ฅ3. I can multiply this
expression by the remaining terms in the
divisor to complete the second column of the table.
๐๐๐๐ โ๐๐๐๐ ๐๐๐๐๐๐๐๐
I know this value, added to โ4๐ฅ๐ฅ3 + 0๐ฅ๐ฅ3, must equal the diagonal sum, 0๐ฅ๐ฅ3.
I need to find an expression that equals 4๐ฅ๐ฅ3 when multiplied by ๐ฅ๐ฅ3. I can multiply this expression by the remaining terms in the divisor to complete the third column of the table and confirm the constant term.
๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐
The expression across the top row of the table is the quotient.
๐๐๐๐๐๐
๐๐๐๐๐๐
๐๐๐๐๐๐
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Homework Helper A Story of FunctionsALGEBRA II
ALG II-M1-HWH-1.3.0-08.2015
โ1๐ฅ๐ฅ 4
๐๐
2015-16 M1 ALGEBRA II
Lesson 4: Comparing MethodsโLong Division, Again?
I repeat the steps in the division algorithm until the remainder has a degree less than that of the divisor.
I need to find an expression that multiplied by ๐ฅ๐ฅ will result in ๐ฅ๐ฅ3, the first term in the dividend. Then I multiply this expression by the terms of the divisor, subtract them from the dividend, and bring down the next term from the dividend.
Lesson 4: Comparing MethodsโLong Division, Again?
1. Is ๐ฅ๐ฅ + 5 a factor of ๐ฅ๐ฅ3 โ 125? Justify your answer using the long division algorithm.
๐๐ + ๐๐ ๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ โ๐๐๐๐๐๐
๐๐๐๐
๐๐ + ๐๐ ๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ โ๐๐๐๐๐๐
โ(๐๐๐๐ +๐๐๐๐๐๐)
โ๐๐๐๐๐๐ +๐๐๐๐
Because long division does not result in a zero remainder, I know that ๐๐ + ๐๐ is not a factor of ๐๐๐๐ โ ๐๐๐๐๐๐.
๐๐๐๐ โ๐๐๐๐
๐๐ + ๐๐ ๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ โ๐๐๐๐๐๐
โ(๐๐๐๐ +๐๐๐๐๐๐)
โ๐๐๐๐๐๐ +๐๐๐๐
โ(โ๐๐๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ โ๐๐๐๐๐๐
๐๐๐๐ โ๐๐๐๐ +๐๐๐๐
๐๐ + ๐๐ ๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ โ๐๐๐๐๐๐
โ(๐๐๐๐ +๐๐๐๐๐๐)
โ๐๐๐๐๐๐ +๐๐๐๐
โ(โ๐๐๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ โ๐๐๐๐๐๐
โ(๐๐๐๐๐๐ +๐๐๐๐๐๐)
โ๐๐๐๐๐๐
I know that, just like with long division of integers, the divisor is a factor of the dividend if the remainder is zero.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 5: Putting It All Together
To rewrite the polynomial, I can use the reverse tabular method or division algorithm (from Lessons 3โ4) to find each quotient and then add the resulting polynomials.
Lesson 5: Putting It All Together
Operations with Polynomials
For Problems 1โ2, quickly determine the first and last terms of each polynomial if it was rewritten in standard form. Then rewrite each expression as a polynomial in standard form.
1. ๐ฅ๐ฅ3โ8๐ฅ๐ฅโ2
+ ๐ฅ๐ฅ3โ๐ฅ๐ฅ2โ10๐ฅ๐ฅโ8๐ฅ๐ฅโ4
The first term is
๐๐๐๐
๐๐+ ๐๐๐๐
๐๐= ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐๐๐,
and the last term is โ๐๐โ๐๐
+ โ๐๐โ๐๐
= ๐๐ + ๐๐ = ๐๐.
Standard form: ๐๐๐๐โ๐๐๐๐โ๐๐
+ ๐๐๐๐โ๐๐๐๐โ๐๐๐๐๐๐โ๐๐๐๐โ๐๐
= ๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ + ๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐
๐๐๐๐ +๐๐๐๐ +๐๐
๐๐ โ ๐๐ ๐๐๐๐ โ๐๐๐๐ โ๐๐๐๐๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ โ๐๐๐๐๐๐
โ(๐๐๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐)
๐๐
๐๐๐๐ +๐๐๐๐ +๐๐
๐๐ โ ๐๐ ๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ +๐๐๐๐
โ(๐๐๐๐๐๐ โ๐๐๐๐)
๐๐๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐)
๐๐
For each quotient, I can find the term with the highest degree by dividing the first term in the numerator by the first term in the denominator. Since the highest-degree terms are like terms, I combine them to find the first term of the polynomial. I can use the same process with the last terms of the quotients to find the last term of the polynomial.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 5: Putting It All Together
I can use the identities (๐๐ โ ๐๐)2 = ๐๐2 โ 2๐๐๐๐ + ๐๐2 and (๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐2 โ ๐๐2 to multiply the polynomials.
I can multiply the first terms in each product and add them to find the first term of the polynomial. I can multiply the constant terms in each product and add them to find the last term.
2. (๐ฅ๐ฅ โ 5)2 + (2๐ฅ๐ฅ โ 1)(2๐ฅ๐ฅ + 1)
The first term is ๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐; the last term is ๐๐๐๐ โ ๐๐ = ๐๐๐๐.
๏ฟฝ๐๐๐๐ + ๐๐(๐๐)(โ๐๐) + (โ๐๐)๐๐๏ฟฝ + ๏ฟฝ(๐๐๐๐)๐๐ โ (๐๐)๐๐๏ฟฝ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๏ฟฝ + ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๏ฟฝ = ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐
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9
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1
Lesson 6: Dividing by ๐ฅ๐ฅ โ ๐๐ and by ๐ฅ๐ฅ + ๐๐
ALGEBRA II
I recognize that I can use an identity to factor the numerator for Problems 4 and 5: ๐ฅ๐ฅ๐๐ โ ๐๐๐๐ = (๐ฅ๐ฅ โ ๐๐)(๐ฅ๐ฅ๐๐โ1 + ๐๐๐ฅ๐ฅ๐๐โ2 + ๐๐2๐ฅ๐ฅ๐๐โ3 +โฏ+ ๐๐๐๐โ2๐ฅ๐ฅ1 + ๐๐๐๐โ1)
I can factor the numerator using the difference of squares identity:
๐ฅ๐ฅ2 โ ๐๐2 = (๐ฅ๐ฅ โ ๐๐)(๐ฅ๐ฅ + ๐๐)
I can rewrite the numerator as (4๐ฅ๐ฅ)3 โ 33. Then I can factor this expression using the difference of cubes identity:
๐ฅ๐ฅ3 โ ๐๐3 = (๐ฅ๐ฅ โ ๐๐)(๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐2).
Lesson 6: Dividing by ๐๐ โ ๐๐ and by ๐๐ + ๐๐
Compute each quotient.
1. 9๐ฅ๐ฅ2โ253๐ฅ๐ฅ+5
(๐๐๐๐)๐๐ โ ๐๐๐๐
๐๐๐๐ + ๐๐=
(๐๐๐๐ โ ๐๐)(๐๐๐๐ + ๐๐)๐๐๐๐ + ๐๐
= ๐๐๐๐ โ ๐๐
2. 64๐ฅ๐ฅ3โ274๐ฅ๐ฅโ3
(๐๐๐๐)๐๐ โ ๐๐๐๐
๐๐๐๐ โ ๐๐=
(๐๐๐๐ โ ๐๐)๏ฟฝ(๐๐๐๐)๐๐ + ๐๐(๐๐๐๐) + ๐๐๐๐๏ฟฝ๐๐๐๐ โ ๐๐
= ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐
3. 8๐ฅ๐ฅ3+11+2x
(๐๐๐๐)๐๐ + ๐๐๐๐
๐๐ + ๐๐๐๐=
(๐๐๐๐ + ๐๐)๏ฟฝ(๐๐๐๐)๐๐ โ ๐๐(๐๐๐๐) + ๐๐๐๐๏ฟฝ๐๐๐๐ + ๐๐
= ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐
4. ๐ฅ๐ฅ9โ1๐ฅ๐ฅโ1
(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ๐๐ โ ๐๐
= ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐ + ๐๐
5. ๐ฅ๐ฅ5โ32๐ฅ๐ฅโ2
๐๐๐๐ โ ๐๐๐๐
๐๐ โ ๐๐=
(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๏ฟฝ๐๐ โ ๐๐
= ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 7: Mental Math
I can rewrite the product 52 โ 28 as the difference of two squares, ๐ฅ๐ฅ2 โ ๐๐2, where ๐ฅ๐ฅ is the arithmetic mean of the factors (numbers being multiplied) and ๐๐ is the positive difference between either factor and ๐ฅ๐ฅ. This means that ๐ฅ๐ฅ = 40 and ๐๐ = 12.
I know that ๐ฅ๐ฅ3 + ๐๐3 = (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ2 โ ๐๐๐ฅ๐ฅ + ๐๐). In this case, ๐ฅ๐ฅ = 12 and ๐๐ = 1.
I know that ๐ฅ๐ฅ2 โ ๐๐2 = (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ โ ๐๐). In this case, ๐ฅ๐ฅ = 11 and ๐๐ = 6.
Lesson 7: Mental Math
Use Polynomial Identities to Perform Arithmetic
1. Using an appropriate polynomial identity, quickly compute the product 52 โ 28. Show each step. Be sureto state your values for ๐ฅ๐ฅ and ๐๐.
๐๐ =๐๐๐๐ + ๐๐๐๐
๐๐= ๐๐๐๐
๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
๐๐๐๐ โ ๐๐๐๐ = (๐๐๐๐ + ๐๐๐๐)(๐๐๐๐ โ ๐๐๐๐) = ๐๐๐๐๐๐โ๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐
2. Rewrite 121 โ 36 as a product of two integers.
๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐ = (๐๐๐๐ + ๐๐)(๐๐๐๐ โ ๐๐) = ๐๐๐๐ โ ๐๐
3. Quickly compute the difference of squares 842 โ 162.
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = (๐๐๐๐ โ ๐๐๐๐)(๐๐๐๐ + ๐๐๐๐) = ๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐
Use Polynomial Identities to Determine if a Number is Prime
4. Is 1729 prime? Use the fact that 123 = 1728 and an identity to support your answer.
The number ๐๐๐๐๐๐๐๐ is not prime because it can be factored as ๐๐๐๐ โ ๐๐๐๐๐๐.
๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐= ๐๐๐๐๐๐ + ๐๐๐๐ = (๐๐๐๐ + ๐๐)๏ฟฝ๐๐๐๐๐๐ โ (๐๐)(๐๐๐๐) + ๐๐๐๐๏ฟฝ = (๐๐๐๐)(๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐) = ๐๐๐๐ โ ๐๐๐๐๐๐
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 7: Mental Math
I need to find a value ๐๐ so 11 is a factor of ๐๐๐๐ โ 1. I can do this by finding a value of ๐๐ so that (๐๐ โ 1) is divisible by 11. One way to do this is to set ๐๐ โ 1 = 11, so that ๐๐ = 12.
I need to rewrite 99,999,951 either as a difference of perfect squares or as a sum or difference of perfect cubes.
5. Show that 99,999,951 is not prime without using a calculator or computer.
Use Polynomial Identities to Determine Divisibility
6. Find a value of ๐๐ so that the expression ๐๐๐๐ โ 1 is always divisible by 11 for any positive integer ๐๐. Explainwhy your value of ๐๐ works for any positive integer ๐๐.
We can factor ๐๐๐๐ โ ๐๐ as follows:
๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐ +โฏ+ ๐๐ + ๐๐๏ฟฝ.
Choose ๐๐ = ๐๐๐๐ so that ๐๐ โ ๐๐ = ๐๐๐๐.
Since ๐๐ โ ๐๐ = ๐๐๐๐, then ๐๐๐๐ โ ๐๐ will have ๐๐๐๐ as a factor
and therefore will always be divisible by ๐๐๐๐.
Note: Any value of ๐๐ where (๐๐ โ 1) is a multiple of 11 will produce a valid solution. Possible values of ๐๐ include 12, 23, 34, and 45.
7. Find a value of ๐๐ so that the expression ๐๐๐๐ โ 1 is divisible by both 3 and 11 for any positive integer ๐๐.Explain why your value of ๐๐ works for any positive integer ๐๐.
We can factor ๐๐๐๐ โ ๐๐ as follows:
๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)(๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐ +โฏ+ ๐๐ + ๐๐).
Choose ๐๐ = ๐๐๐๐ so that ๐๐ โ ๐๐ = ๐๐๐๐.
Since ๐๐ โ ๐๐ = ๐๐๐๐, then ๐๐๐๐ โ ๐๐ will have ๐๐๐๐ as a factor, which is a multiple of both ๐๐ and ๐๐๐๐. Therefore, ๐๐๐๐ โ ๐๐ will always be divisible by ๐๐ and ๐๐๐๐.
Note: Any value of ๐๐ where (๐๐ โ 1) is a multiple of 33 will produce a valid solution. Possible values of ๐๐ include 34, 67, and 100
๐๐๐๐,๐๐๐๐๐๐,๐๐๐๐๐๐ = ๐๐๐๐๐๐,๐๐๐๐๐๐,๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐ = ๏ฟฝ๐๐๐๐๐๐๏ฟฝ๐๐ โ ๐๐๐๐ = ๏ฟฝ๐๐๐๐๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๏ฟฝ = ๐๐๐๐,๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 8: The Power of AlgebraโFinding Primes
Lesson 8: The Power of AlgebraโFinding Primes
Apply Polynomial Identities to Factor Composite Numbers
1. Factor 512 โ 1 in three different ways: using the identity (๐ฅ๐ฅ๐๐ โ ๐๐๐๐)(๐ฅ๐ฅ๐๐โ1 + ๐๐๐ฅ๐ฅ๐๐โ2 +โฏ+ ๐๐๐๐โ1), thedifference of squares identity, and the difference of cubes identity.
i. ๐๐๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + โฏ+ ๐๐๐๐ + ๐๐๏ฟฝ
ii. ๐๐๐๐๐๐ โ ๐๐ = ๏ฟฝ๐๐๐๐๏ฟฝ๐๐ โ ๐๐๐๐ = (๐๐๐๐ โ ๐๐)(๐๐๐๐ + ๐๐)
iii. ๐๐๐๐๐๐ โ ๐๐ = ๏ฟฝ๐๐๐๐๏ฟฝ๐๐ โ ๐๐๐๐
= ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ ๏ฟฝ๏ฟฝ๐๐๐๐๏ฟฝ๐๐ + ๐๐๏ฟฝ๐๐๐๐๏ฟฝ + ๐๐๐๐๏ฟฝ= ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ(๐๐๐๐ + ๐๐๐๐ + ๐๐)
Apply Polynomial Identities to Analyze Divisibility
2. Explain why if ๐๐ is odd, the number 3๐๐ + 1 will always be divisible by 4.
For any odd value of ๐๐,
๐๐๐๐ + ๐๐ = ๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐)๏ฟฝ๐๐๐๐โ๐๐ โ (๐๐)๐๐๐๐โ๐๐ + ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐๐โ๐๐ โโฏ+ ๏ฟฝ๐๐๐๐โ๐๐๏ฟฝ๐๐๐๐ โ ๏ฟฝ๐๐๐๐โ๐๐๏ฟฝ๐๐ + ๐๐๐๐โ๐๐๏ฟฝ = ๐๐๏ฟฝ๐๐๐๐โ๐๐ โ (๐๐)๐๐๐๐โ๐๐ + ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐๐โ๐๐ โโฏ+ ๏ฟฝ๐๐๐๐โ๐๐๏ฟฝ๐๐๐๐ โ ๏ฟฝ๐๐๐๐โ๐๐๏ฟฝ๐๐ + ๐๐๐๐โ๐๐๏ฟฝ.
Since ๐๐ is a factor of ๐๐๐๐ + ๐๐ for any odd value of ๐๐, the number ๐๐๐๐ + ๐๐ is divisible by ๐๐.
I know that 512 โ 1 can be written as a difference of squares because it can be written in the form ๐ฅ๐ฅ2๐๐ โ ๐๐2๐๐, where ๐ฅ๐ฅ = 5, ๐๐ = 1 and ๐๐ = 6. I can then apply the difference of squares identity:
๐ฅ๐ฅ2๐๐ โ ๐๐2๐๐ = (๐ฅ๐ฅ๐๐ โ ๐๐๐๐)(๐ฅ๐ฅ๐๐ + ๐๐๐๐).
For odd numbers ๐๐, I can use the identity ๐ฅ๐ฅ๐๐ + ๐๐๐๐ = (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ๐๐โ1 โ ๐๐๐ฅ๐ฅ๐๐โ2 + ๐๐2๐ฅ๐ฅ๐๐โ3 โ โฏ+ ๐๐๐๐โ1).
I know that 512 โ 1 can be written as a difference of cubes because it can be written in the form ๐ฅ๐ฅ3๐๐ โ ๐๐3๐๐, where ๐ฅ๐ฅ = 5, ๐๐ = 1 and ๐๐ = 4. I can then apply the difference of cubes identity:
๐ฅ๐ฅ3๐๐ โ ๐๐3๐๐ = (๐ฅ๐ฅ๐๐ โ ๐๐๐๐)(๐ฅ๐ฅ2๐๐ + ๐๐๐๐๐ฅ๐ฅ๐๐
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Homework Helper A Story of Functions
๐๐2๐๐+ )
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 8: The Power of AlgebraโFinding Primes
3. If ๐๐ is a composite number, explain why 4๐๐ โ 1 is never prime.Since ๐๐ is composite, there are integers ๐๐ and ๐๐larger than ๐๐ so that ๐๐ = ๐๐๐๐. Then
๐๐๐๐ โ ๐๐ = ๐๐๐๐๐๐ โ ๐๐ = (๐๐๐๐ โ ๐๐)๏ฟฝ(๐๐๐๐)๐๐โ๐๐ + (๐๐๐๐)๐๐โ๐๐ + (๐๐๐๐)๐๐โ๐๐ + โฏ+ (๐๐๐๐)๐๐ + ๐๐๏ฟฝ.
Since ๐๐ > ๐๐, the expression ๐๐๐๐ โ ๐๐ must be an integer greater than or equal to ๐๐๐๐, so ๐๐๐๐๐๐ โ ๐๐ has a factor other than ๐๐. Therefore, the number ๐๐๐๐ โ ๐๐ is composite.
Apply the Difference of Squares Identity to Rewrite Numbers
4. Express the numbers from 31 to 40 as the difference of two squares, if possible. The first four have beendone for you.
Number Factorization Difference of two squares
31 31 = 31 โ 1 = (16 + 15)(16โ 15)
162 โ 152 = 256 โ 225 = 31
32 32 = 8 โ 4 = (6 + 2)(6 โ 2)
62 โ 22 = 36 โ 4 = 32
33 33 = 11 โ 3 = (7 + 4)(7 โ 4)
72 โ 42 = 49 โ 16 = 33
34 Canโt be done.
35 ๐๐๐๐ = ๐๐ โ ๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐)
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = ๐๐๐๐
36 ๐๐๐๐ = ๐๐ โ ๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐)
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = ๐๐๐๐
37 ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = (๐๐๐๐ + ๐๐๐๐)(๐๐๐๐ โ ๐๐๐๐)
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐
38 Canโt be done
39 ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐)
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
40 ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐)
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = ๐๐๐๐
I need to write each number as a product of two whole numbers. I can rewrite the product as a difference of two squares using the method we learned in Lesson 7.
If I cannot factor the number into two positive integers with an even sum, the number cannot be written as a difference of squares because the mean would be a fraction.
I can use the identity ๐ฅ๐ฅ๐๐ โ 1 = (๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ๐๐โ1 + ๐ฅ๐ฅ๐๐โ2 + ๐ฅ๐ฅ๐๐โ3 + โฏ+ ๐ฅ๐ฅ + 1).
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ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 9: Radicals and Conjugates
To write the denominator as an integer, I can multiply the numerator and denominator by a common factor that makes the radicand in the denominator a perfect cube.
Lesson 9: Radicals and Conjugates
Convert Expressions to Simplest Radical Form
Express each of the following as a rational expression or in simplest radical form. Assume that the symbol ๐๐ represents a positive number. Remember that a simplified radical expression has a denominator that is an integer.
1. โ5๐ฅ๐ฅ๏ฟฝโ10 โ โ2๐ฅ๐ฅ๏ฟฝ
โ๐๐๐๐ โ โ๐๐๐๐ โ โ๐๐๐๐ โ โ๐๐๐๐ = โ๐๐๐๐๐๐ โ ๏ฟฝ๐๐๐๐๐๐๐๐
= โ๐๐๐๐ โ ๐๐๐๐ โ ๏ฟฝ๐๐๐๐ โ ๐๐๐๐
= โ๐๐๐๐ โ โ๐๐๐๐ โ โ๐๐๐๐ โ ๏ฟฝ๐๐๐๐ = ๐๐โ๐๐๐๐ โ ๐๐โ๐๐๐๐
2. ๏ฟฝ 2
27
3 + โ243 โ ๏ฟฝ1
4
3
I can distribute the โ5๐ฅ๐ฅ to each term in the parentheses by multiplying the radicands.
Once I distribute the โ5๐ฅ๐ฅ term to each term in the parentheses, I can factor each radicand so that one factor is a perfect square. The perfect square factors can be taken outside the radical.
I know that I can
rewrite ๏ฟฝ๐๐๐๐
3 as โ๐๐3
โ๐๐3 ,
which will allow me to simplify the numerator and denominator separately.
๏ฟฝ ๐๐๐๐๐๐
๐๐+ โ๐๐๐๐๐๐ โ ๏ฟฝ
๐๐๐๐
๐๐=
โ๐๐๐๐
โ๐๐๐๐๐๐ + โ๐๐ โ ๐๐๐๐ โ โ๐๐๐๐
โ๐๐๐๐
=โ๐๐๐๐
โ๐๐๐๐๐๐ + โ๐๐๐๐ โ โ๐๐๐๐ โ โ๐๐ โ ๐๐๐๐
โ๐๐ โ ๐๐๐๐
=โ๐๐๐๐
๐๐+ ๐๐โ๐๐๐๐ โ
โ๐๐๐๐
โ๐๐๐๐
=โ๐๐๐๐
๐๐+ ๐๐โ๐๐๐๐ โ
โ๐๐๐๐
๐๐
= ๐๐โ๐๐๐๐
๐๐+ ๐๐โ๐๐๐๐ โ
๐๐โ๐๐๐๐
๐๐
= ๐๐โ๐๐๐๐ โโ๐๐๐๐
๐๐
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 9: Radicals and Conjugates
I can convert the denominator to an integer by multiplying both the numerator and denominator of the fraction by the radical conjugate.
Since 2โ3 โ 4โ5 is the expression, 2โ3 + 4โ5 is its radical conjugate.
I know that for any positive numbers ๐๐ and ๐๐,
๏ฟฝโ๐๐ + โ๐๐๏ฟฝ๏ฟฝโ๐๐ โ โ๐๐๏ฟฝ = ๏ฟฝโ๐๐๏ฟฝ2โ ๏ฟฝโ๐๐๏ฟฝ
2. In this
case, ๐๐ = ๏ฟฝ2โ3๏ฟฝ2
= 12 and ๐๐ = ๏ฟฝ4โ5๏ฟฝ2
= 80.
Simplify Radical Quotients
Simplify each of the following quotients as far as possible.
3. โ123 โ โ33
โ33
4. 5โ2โโ32โ3โ4โ5
๏ฟฝโ๐๐๐๐๐๐ โ โ๐๐๐๐ ๏ฟฝโ๐๐๐๐ =
โ๐๐๐๐๐๐
โ๐๐๐๐ โโ๐๐๐๐
โ๐๐๐๐
= ๏ฟฝ๐๐๐๐๐๐
๐๐โ ๏ฟฝ๐๐
๐๐๐๐
= โ๐๐๐๐ โ ๐๐
I remember that an expression is in simplified radical form if it has no terms with an exponent greater than or equal to the index of the radical and if it does not have a radical in the denominator.
๐๐โ๐๐ โ โ๐๐๐๐โ๐๐ โ ๐๐โ๐๐
โ๐๐โ๐๐ + ๐๐โ๐๐๐๐โ๐๐ + ๐๐โ๐๐
=๏ฟฝ๐๐โ๐๐ โ โ๐๐๏ฟฝ โ ๏ฟฝ๐๐โ๐๐ + ๐๐โ๐๐๏ฟฝ๏ฟฝ๐๐โ๐๐ โ ๐๐โ๐๐๏ฟฝ โ ๏ฟฝ๐๐โ๐๐ + ๐๐โ๐๐๏ฟฝ
=๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐๐๐ โ ๐๐โ๐๐ โ ๐๐โ๐๐๐๐
(๐๐โ๐๐)๐๐ โ (๐๐โ๐๐)๐๐
=๐๐๐๐โ๐๐ + ๐๐๐๐โ๐๐๐๐ โ ๐๐(๐๐) โ ๐๐โ๐๐๐๐
๐๐๐๐ โ ๐๐๐๐
=๐๐๏ฟฝ๐๐โ๐๐ + ๐๐๐๐โ๐๐๐๐ โ ๐๐ โ ๐๐โ๐๐๐๐๏ฟฝ
โ๐๐๐๐
=๐๐โ๐๐ + ๐๐๐๐โ๐๐๐๐ โ ๐๐ โ ๐๐โ๐๐๐๐
โ๐๐๐๐
=๐๐ + ๐๐โ๐๐๐๐ โ ๐๐โ๐๐ โ ๐๐๐๐โ๐๐๐๐
๐๐๐๐
ยฉ 2015 Great Minds eureka-math.org
16
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16
Lesson 10: The Power of AlgebraโFinding Pythagorean Triples
M1 ALGEBRA II
I see that the constant term, 25, is a perfect square and that the coefficient of the linear ๐ฅ๐ฅ term is 10, which is twice the square root of 25. That reminds me of the identity: (๐ฅ๐ฅ โ ๐๐)2 = ๐ฅ๐ฅ2 โ 2๐๐๐ฅ๐ฅ + ๐๐2.
I need to write the expression in the form (๐๐ + ๐๐)(๐๐ โ ๐๐). Since both the ๐ฆ๐ฆ-term and the constant, 1, switch signs between the factors, I know that the ๐๐ component of the identity is ๐ฆ๐ฆ + 1.
Lesson 10: The Power of AlgebraโFinding Pythagorean Triples
Use the difference of squares identity ๐๐๐๐ โ ๐๐๐๐ = (๐๐ โ ๐๐)(๐๐ + ๐๐) to simplify algebraic expressions.
1. Use the difference of squares identity to find (๐ฅ๐ฅ โ ๐ฆ๐ฆ โ 1)(๐ฅ๐ฅ + ๐ฆ๐ฆ + 1).
Use the difference of squares identity to factor algebraic expressions.
2. Use the difference of two squares identity to factor the expression (๐ฅ๐ฅ + ๐ฆ๐ฆ)(๐ฅ๐ฅ โ ๐ฆ๐ฆ) โ 10๐ฅ๐ฅ + 25.
(๐๐ โ ๐๐ โ ๐๐)(๐๐ + ๐๐ + ๐๐) = ๏ฟฝ๐๐ โ (๐๐ + ๐๐)๏ฟฝ๏ฟฝ๐๐ + (๐๐ + ๐๐)๏ฟฝ = ๐๐๐๐ โ (๐๐ + ๐๐)๐๐ = ๐๐๐๐ โ ๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐
(๐๐ + ๐๐)(๐๐ โ ๐๐) โ ๐๐๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๏ฟฝ โ ๐๐๐๐ = (๐๐ โ ๐๐)๐๐ โ ๐๐๐๐ = ๏ฟฝ(๐๐ โ ๐๐) โ ๐๐๏ฟฝ๏ฟฝ(๐๐ โ ๐๐) + ๐๐๏ฟฝ
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17
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16
Lesson 10: The Power of AlgebraโFinding Pythagorean Triples
M1 ALGEBRA II
I know ๐ฅ๐ฅ = 4 and ๐ฆ๐ฆ = 1 because ๐ฅ๐ฅ๐ฆ๐ฆ = 4 and ๐ฅ๐ฅ > ๐ฆ๐ฆ.
I need to try all the cases where ๐ฅ๐ฅ๐ฆ๐ฆ = 18 and ๐ฅ๐ฅ > ๐ฆ๐ฆ.
Apply the difference of squares identity to find Pythagorean triples.
3. Prove that the Pythagorean triple (15, 8, 17) can be found by choosing a pair of integers ๐ฅ๐ฅ and ๐ฆ๐ฆ with๐ฅ๐ฅ > ๐ฆ๐ฆ and computing (๐ฅ๐ฅ2 โ ๐ฆ๐ฆ2, 2๐ฅ๐ฅ๐ฆ๐ฆ, ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2).
4. Prove that the Pythagorean triple (15, 36, 39) cannot be found by choosing a pair of integers ๐ฅ๐ฅ and ๐ฆ๐ฆwith ๐ฅ๐ฅ > ๐ฆ๐ฆ and computing (๐ฅ๐ฅ2 โ ๐ฆ๐ฆ2, 2๐ฅ๐ฅ๐ฆ๐ฆ, ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2).
โข If ๐๐ = ๐๐๐๐ and ๐๐ = ๐๐, then ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐ = ๐๐๐๐๐๐. So this combination does notproduce a number in the triple.
โข If ๐๐ = ๐๐ and ๐๐ = ๐๐, then ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = ๐๐๐๐. So this combination does not producea number in the triple.
โข If ๐๐ = ๐๐ and ๐๐ = ๐๐, then ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = ๐๐๐๐. So this combination does not producea number in the triple.
There are no other integer values of ๐๐ and ๐๐ that satisfy ๐๐๐๐ = ๐๐๐๐ and ๐๐ > ๐๐. Therefore, the triple (๐๐๐๐,๐๐๐๐,๐๐๐๐) cannot be found using the method described.
We want ๐๐๐๐๐๐ = ๐๐, so ๐๐๐๐ = ๐๐, so we can set ๐๐ = ๐๐ and ๐๐ = ๐๐.
This means that ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ and๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐.
Therefore, the values ๐๐ = ๐๐ and ๐๐ = ๐๐ generate thePythagorean triple
๏ฟฝ๐๐๐๐ โ ๐๐๐๐,๐๐๐๐๐๐,๐๐๐๐ + ๐๐๐๐๏ฟฝ = (๐๐๐๐,๐๐,๐๐๐๐).
Since ๐๐๐๐ is the only even number in (๐๐๐๐,๐๐๐๐,๐๐๐๐), we must have ๐๐๐๐๐๐ = ๐๐๐๐, so ๐๐๐๐ = ๐๐๐๐.
ยฉ 2015 Great Minds eureka-math.org
18
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 11: The Special Role of Zero in Factoring
Lesson 11: The Special Role of Zero in Factoring
Find Solutions to Polynomial Equations
Find all solutions to the given equations.
1. 2๐ฅ๐ฅ(๐ฅ๐ฅ + 1)2(3๐ฅ๐ฅ โ 4) = 0
๐๐๐๐ = ๐๐ or (๐๐ + ๐๐) = ๐๐ or (๐๐๐๐ โ ๐๐) = ๐๐
Solutions: ๐๐, โ๐๐, ๐๐๐๐
2. ๐ฅ๐ฅ(๐ฅ๐ฅ2 โ 25)(๐ฅ๐ฅ2 โ 1) = 0
๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐
๐๐ = ๐๐ or (๐๐ โ ๐๐) = ๐๐ or (๐๐ + ๐๐) = ๐๐ or (๐๐ โ ๐๐) = ๐๐ or (๐๐ + ๐๐) = ๐๐
Solutions: ๐๐, ๐๐, โ๐๐, ๐๐, โ๐๐
3. (๐๐ + ๐๐)๐๐ = (๐๐๐๐ โ 6)๐๐
(๐๐ + ๐๐)๐๐ โ (๐๐๐๐ โ ๐๐)๐๐ = ๐๐
๏ฟฝ(๐๐ + ๐๐) + (๐๐๐๐ โ ๐๐)๏ฟฝ๏ฟฝ(๐๐ + ๐๐) โ (๐๐๐๐ โ ๐๐)๏ฟฝ = ๐๐
(๐๐๐๐ โ ๐๐)(โ๐๐๐๐+ ๐๐) = ๐๐
๐๐๐๐ โ ๐๐ = ๐๐ or โ๐๐๐๐ + ๐๐ = ๐๐
Solutions: ๐๐๐๐, ๐๐๐๐
Determine Zeros and Multiplicity for Polynomial Functions
4. Find the zeros with multiplicity for the function ๐๐(๐ฅ๐ฅ) = (๐ฅ๐ฅ3 โ 1)(๐ฅ๐ฅ4 โ 9๐ฅ๐ฅ2).The number of times a solution appears as a factor in a polynomial function is its multiplicity.
๐๐ = (๐๐ โ ๐๐)(๐๐๐๐ + ๐๐ + ๐๐)๐๐๐๐(๐๐๐๐ โ ๐๐)
๐๐ = (๐๐ โ ๐๐)(๐๐๐๐ + ๐๐ + ๐๐)๐๐๐๐(๐๐ + ๐๐)(๐๐ โ ๐๐)
Then ๐๐ is a zero of multiplicity ๐๐, ๐๐ is a zero of multiplicity ๐๐, โ๐๐ is a zero of multiplicity ๐๐, and ๐๐ is a zero of multiplicity ๐๐.
To find all the solutions, I need to set each factor equal to zero and solve the resulting equations.
I factored the expression (๐ฅ๐ฅ3 โ 1) using the difference of cubes pattern. Since (๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 1) does not factor, it does not contribute any zeros of the function ๐๐.
I recognize the expression on the left side of the equation as the difference of two perfect squares. I know that ๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐). In this case, ๐๐ = (๐ฅ๐ฅ + 3) and ๐๐ = (3๐ฅ๐ฅ โ 6).
I collected like terms to rewrite the factors.
The exponent 2 means that the equation has a repeated factor, which corresponds to a repeated solution.
ยฉ 2015 Great Minds eureka-math.org
19
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 11: The Special Role of Zero in Factoring
Construct a Polynomial Function That Has a Specified Set of Zeros with Stated Multiplicity
5. Find two different polynomial functions that have a zero at 1 of multiplicity 3 and a zero at โ2 ofmultiplicity 2.
๐๐(๐๐) = (๐๐ โ ๐๐)๐๐(๐๐ + ๐๐)๐๐
๐๐(๐๐) = ๐๐(๐๐ โ ๐๐)๐๐(๐๐ + ๐๐)๐๐
Compare the Remainder of ๐๐(๐๐) รท (๐๐ โ ๐๐) with ๐๐(๐๐)
6. Consider the polynomial function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6.a. Divide ๐๐ by the divisor (๐ฅ๐ฅ โ 2), and rewrite in the form ๐๐(๐ฅ๐ฅ) = (divisor)(quotient) + remainder.
Either the division algorithm or the reverse tabular method can be used to find the quotient andremainder.
๐๐(๐๐) = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ + ๐๐
b. Evaluate ๐๐(2).
๐๐(๐๐) = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ + ๐๐๐๐(๐๐) = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐ โ ๐๐ + ๐๐๏ฟฝ + ๐๐๐๐(๐๐) = ๐๐ + ๐๐๐๐(๐๐) = ๐๐
๐๐๐๐ +๐๐๐๐ +๐๐
๐๐ โ ๐๐ ๐๐๐๐ +๐๐๐๐ +๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ +๐๐
โ(๐๐๐๐๐๐ โ๐๐๐๐)
๐๐๐๐ โ๐๐
โ(๐๐๐๐ โ๐๐๐๐)
๐๐
I need to include the linear factor (๐ฅ๐ฅ โ 1) three times and the linear factor (๐ฅ๐ฅ + 2) twice in the statement of each function.
I used the division algorithm we learned in Lesson 4 to find the quotient and remainder.
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20
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 12: Overcoming Obstacles In Factoring
๐๐๐๐ โ ๐๐๐๐ = โ๐๐ ๐๐๐๐ โ ๐๐๐๐ + (โ๐๐)๐๐ = โ๐๐ + (โ๐๐)๐๐
๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = โ๐๐ + ๐๐๐๐ (๐๐ โ ๐๐)๐๐ = ๐๐๐๐
๐๐ โ ๐๐ = ยฑโ๐๐๐๐
Lesson 12: Overcoming Obstacles In Factoring
Solving Problems by Completing the Square
Solve each of the following equations by completing the square.
1. ๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ + 3 = 0
The solutions are ๐๐ + โ๐๐๐๐ and ๐๐ โ โ๐๐๐๐.
2. 2๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ = โ1
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = โ๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐+ ๐๐ = โ๐๐+ ๐๐
(๐๐๐๐ โ ๐๐)๐๐ = ๐๐ ๐๐๐๐ โ ๐๐ = ยฑโ๐๐
The solutions are ๐๐+โ๐๐๐๐
and ๐๐โโ๐๐๐๐
.
3. ๐ฅ๐ฅ4 โ 6๐ฅ๐ฅ2 + 4 = 0
๐๐๐๐ โ ๐๐๐๐๐๐ = โ๐๐ ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐ = โ๐๐+ ๐๐
(๐๐๐๐ โ ๐๐)๐๐ = ๐๐ ๐๐๐๐ โ ๐๐ = ยฑโ๐๐
๐๐๐๐ = ๐๐ ยฑ โ๐๐
The solutions are ๏ฟฝ๐๐ + โ๐๐,๏ฟฝ๐๐ โ โ๐๐,โ๏ฟฝ๐๐ + โ๐๐, and โ๏ฟฝ๐๐ โ โ๐๐.
I squared half the linear coefficient and added the result to both sides of the equation.
I recognize that ๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ + 16 fits the pattern for the square of a difference: ๐ฅ๐ฅ2 โ 2๐๐๐ฅ๐ฅ + ๐๐2 = (๐ฅ๐ฅ โ ๐๐)2
I multiplied both sides of the equation by 2 so the quadratic term would be a perfect square. Then I can complete the square using the reverse tabular method, shown to the right.
๐๐๐๐ โ๐๐
๐๐๐๐ ๐๐๐๐๐๐ โ๐๐๐๐
โ๐๐ โ๐๐๐๐ ๐๐
ยฉ 2015 Great Minds eureka-math.org
21
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 12: Overcoming Obstacles In Factoring
Solving Problems by Using the Quadratic Formula
Solve the equation by using the quadratic formula.
4. (๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3)(๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + 2) = 0
๐๐ = ๐๐ยฑ๏ฟฝ(โ๐๐)๐๐โ๐๐(๐๐)(๐๐)๐๐(๐๐)
or ๐๐ = ๐๐ยฑ๏ฟฝ(โ๐๐)๐๐โ๐๐(๐๐)(๐๐)๐๐(๐๐)
The solutions are ๐๐+โ๐๐๐๐๐๐
, ๐๐โโ๐๐๐๐๐๐
, ๐๐+โ๐๐๐๐๐๐
, and ๐๐โโ๐๐๐๐๐๐
.
Solving Polynomial Equations by Factoring
Use factoring to solve each equation.
5. 6๐ฅ๐ฅ2 + 7๐ฅ๐ฅ โ 3 = 0
๐๐๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ โ ๐๐ = ๐๐ ๐๐๐๐(๐๐๐๐ + ๐๐) โ ๐๐(๐๐๐๐ + ๐๐) = ๐๐
(๐๐๐๐ โ ๐๐)(๐๐๐๐ + ๐๐) = ๐๐
The solutions are โ๐๐๐๐
and ๐๐๐๐.
6. (2๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ โ 4) = (2๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 3)
(๐๐๐๐ โ ๐๐)(๐๐ โ ๐๐) โ (๐๐๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐(๐๐๐๐ โ ๐๐)๏ฟฝ(๐๐ โ ๐๐) โ (๐๐ + ๐๐)๏ฟฝ = ๐๐
(๐๐๐๐ โ ๐๐)(โ๐๐) = ๐๐
The only solution is ๐๐๐๐.
7. (๐ฅ๐ฅ2 + 5)2 โ 3 = 0
The solutions are ๏ฟฝโ๐๐ + โ๐๐, โ๏ฟฝโ๐๐ + โ๐๐,๏ฟฝโ๐๐ โ โ๐๐, and โ๏ฟฝโ๐๐โ โ๐๐.
I can find the solutions by setting each factor equal to zero. I can solve the resulting quadratic equations using the quadratic formula: If ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, then
๐ฅ๐ฅ = โ๐๐ยฑโ๐๐2โ4๐๐๐๐2๐๐
.
I split the linear term into two terms whose coefficients multiply to the product of the leading coefficient and the constant (9 โ โ2) = (6 โ โ3). Then I can factor the resulting quadratic expression by grouping.
I factored (2๐ฅ๐ฅ โ 1) from both expressions.
I factored the left side of the equation as a difference of two perfect squares: ๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐).
๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ โ โ๐๐๏ฟฝ ๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ + โ๐๐๏ฟฝ = ๐๐
๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ โ โ๐๐๏ฟฝ = ๐๐ or ๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ + โ๐๐๏ฟฝ = ๐๐
๐๐๐๐ = โ๐๐ + โ๐๐ or ๐๐๐๐ = โ๐๐ โ โ๐๐
ยฉ 2015 Great Minds eureka-math.org
22
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 ALGEBRA II
Lesson 13: Mastering Factoring
Lesson 13: Mastering Factoring
Factor Polynomials
1. If possible, factor the following polynomials over the real numbers.
a. 9๐ฅ๐ฅ2๐ฆ๐ฆ2 + 6๐ฅ๐ฅ๐ฆ๐ฆ โ 8
๐๐(๐๐๐๐)๐๐ + ๐๐(๐๐๐๐) โ ๐๐ = ๐๐(๐๐๐๐)๐๐ + ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐= ๐๐๐๐๐๐(๐๐๐๐๐๐ + ๐๐) โ ๐๐(๐๐๐๐๐๐+ ๐๐) = (๐๐๐๐๐๐ โ ๐๐)(๐๐๐๐๐๐ + ๐๐)
b. 18๐ฅ๐ฅ3 โ 50๐ฅ๐ฅ๐ฆ๐ฆ4๐ง๐ง6
๐๐๐๐๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐๐๐๏ฟฝ = ๐๐๐๐ ๏ฟฝ(๐๐๐๐)๐๐ โ ๏ฟฝ๐๐๐๐๐๐๐๐๐๐๏ฟฝ๐๐๏ฟฝ= ๐๐๐๐๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๐๐๐๐๐๐๐๐๏ฟฝ
c. 3๐ฅ๐ฅ5 โ 81๐ฅ๐ฅ2
๐๐๐๐๐๐๏ฟฝ๐๐๐๐ โ ๐๐๐๐๏ฟฝ = ๐๐๐๐๐๐(๐๐ โ ๐๐)(๐๐๐๐ + ๐๐๐๐ + ๐๐)
d. 16๐ฅ๐ฅ4 + 25๐ฆ๐ฆ2
Cannot be factored over the real numbers
Since the greatest common factor (GCF) among the terms is 1 and there are 3 terms, I split the middle term so its coefficients multiplied to the product of the leading coefficient and the constant: 12(โ6) = 9(โ8). I can then factor the polynomial by grouping just as we did in Lesson 12.
After factoring out the GCF, 2๐ฅ๐ฅ, I recognized the resulting expression was a difference of two perfect squares because the coefficients of the terms were perfect squares, and the exponents were even.
This is the sum of two perfect squares, and I cannot factor the sum of squares over the real numbers.
After factoring out the GCF, I recognize the factor ๐ฅ๐ฅ3 โ 72 as a difference of cubes.
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Homework Helper A Story of Functions M1
ALG II-M1-HWH-1.3.0-08.2015
2015-16 ALGEBRA II
Lesson 13: Mastering Factoring
e. 2๐ฅ๐ฅ4 โ 50
๐๐๏ฟฝ๐๐๐๐ โ ๐๐๐๐๏ฟฝ = ๐๐๏ฟฝ๏ฟฝ๐๐๐๐๏ฟฝ๐๐ โ ๐๐๐๐๏ฟฝ= ๐๐๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ = ๐๐๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐ โ โ๐๐๏ฟฝ๏ฟฝ๐๐ + โ๐๐๏ฟฝ
2.
a. Factor ๐ฆ๐ฆ6 โ 1 first as a difference of cubes, and then factor completely: (๐ฆ๐ฆ2)3 โ 13.
๐๐๐๐ โ ๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๏ฟฝ= (๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐๐๐ + ๐๐๐๐ + ๐๐)
b. Factor ๐ฆ๐ฆ6 โ 1 first as a difference of squares, and then factor completely: (๐ฆ๐ฆ3)2 โ 12.
๐๐๐๐ โ ๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ= (๐๐ โ ๐๐)(๐๐๐๐ + ๐๐ + ๐๐)(๐๐ + ๐๐)(๐๐๐๐ โ ๐๐ โ ๐๐)
c. Use your results from parts (a) and (b) to factor ๐ฆ๐ฆ4 + ๐ฆ๐ฆ2 + 1.
From part (a), we know ๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)(๐๐ + ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ.From part (b), we know ๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ(๐๐ + ๐๐)๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ, which is equivalentto ๐๐๐๐ โ ๐๐ = (๐๐ โ ๐๐)(๐๐ + ๐๐)๏ฟฝ๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ.
It follows that ๐๐๐๐ + ๐๐๐๐ + ๐๐ = (๐๐๐๐ + ๐๐ + ๐๐)( ๐๐๐๐ โ ๐๐ โ ๐๐).
When I factor the expression as a difference of cubes, the first expression can be factored as a difference of squares.
When I factor the expression as a difference of squares, the first expression can be factored as a difference of cubes and the second as the sum of cubes.
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Homework Helper A Story of Functions M1
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 14: Graphing Factored Polynomials
Lesson 14: Graphing Factored Polynomials
Steps for Creating Sketches of Factored Polynomials
First, determine the zeros of the polynomial from the factors. These values are the ๐ฅ๐ฅ-intercepts of the graph of the function. Mark these values on the ๐ฅ๐ฅ-axis of a coordinate grid.
Evaluate the function at ๐ฅ๐ฅ-values between the zeros to determine whether the graph of the function is negative or positive between the zeros. If the output is positive, the graph lies above the ๐ฅ๐ฅ-axis. If the output is negative, the graph lies below the ๐ฅ๐ฅ-axis.
Determining the Number of ๐๐-intercepts and Relative Maxima and Minima for Graphs of Polynomials
1. For the function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ5 โ ๐ฅ๐ฅ3 โ ๐ฅ๐ฅ2 + 1, identify the largest possible number of ๐ฅ๐ฅ-intercepts and thelargest possible number of relative maxima and minima based on the degree of the polynomial. Then usea calculator or graphing utility to graph the function and find the actual number of ๐ฅ๐ฅ-intercepts andrelative maxima and minima.
The largest number of possible ๐๐-intercepts is ๐๐, and the largest possible number of relative maximaand minima is ๐๐.
The graph of ๐๐ has two ๐๐-intercepts, one relative maximum point, and one relative minimum point.
I know the largest possible number of ๐ฅ๐ฅ-intercepts is equal to the degree of the polynomial, and the largest possible number of relative maxima and relative minima is equal to one less than the degree of the polynomial.
I know (1, 0) is a relative minimum point because the output values for inputs immediately to the right and left of 1 are greater than 0.
I know (0, 1) is a relative maximum point because the output values for inputs immediately greater than or less than 0 are less than 1.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 14: Graphing Factored Polynomials
Sketching a Graph from Factored Polynomials
2. Sketch a graph of the function ๐๐(๐ฅ๐ฅ) = 2(๐ฅ๐ฅ + 4)(๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 2) by finding the zeros and determining thesign of the values of the function between zeros.
The zeros are โ๐๐,๐๐, and โ๐๐.
For ๐๐ < โ๐๐: ๐๐(โ๐๐) = โ๐๐๐๐, so the graph of ๐๐ is belowthe ๐๐-axis for ๐๐ < โ๐๐.
For โ๐๐ < ๐๐ < โ๐๐: ๐๐(โ๐๐) = ๐๐, so the graph of ๐๐ isabove the ๐๐-axis for โ๐๐ < ๐๐ < โ๐๐.
For โ๐๐ < ๐๐ < ๐๐: ๐๐(๐๐) = โ๐๐๐๐, so the graph of ๐๐ isbelow the ๐๐-axis for โ๐๐ < ๐๐ < ๐๐.
For ๐๐ > ๐๐: ๐๐(๐๐) = ๐๐๐๐, so the graph is above the ๐๐-axis for ๐๐ > ๐๐.
Since โ4 is the zero with the least value, I know the graph of ๐๐ will not intersect the ๐ฅ๐ฅ-axis for inputs less than โ4, which means that since ๐๐(โ5) is negative, all outputs for ๐ฅ๐ฅ < โ4 will also be negative, and this part of the graph will be below the ๐ฅ๐ฅ-axis.
Since there are no zeros for โ4 < ๐ฅ๐ฅ < โ2, I know the graph of ๐๐ will not intersect the ๐ฅ๐ฅ-axis for inputs between โ4 and โ2. Then, since ๐๐(โ3) is positive, all outputs for โ4 < ๐ฅ๐ฅ < โ2 will also be positive, and this part of the graph will be above the ๐ฅ๐ฅ-axis.
If I substitute input values between the zeros, I can determine from the resulting outputs if the graph of ๐๐ will be above or below the ๐ฅ๐ฅ-axis for the regions between the zeros.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 14: Graphing Factored Polynomials
3. Sketch a graph of the function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ4 โ 4๐ฅ๐ฅ3 + 3๐ฅ๐ฅ2 + 4๐ฅ๐ฅ โ 4 by determining the sign of the values ofthe function between zeros โ1, 1, and 2.
4. A function ๐๐ has zeros at โ1, 2, and 4. We know that ๐๐(โ2) and ๐๐(0) are positive, while ๐๐(3) and ๐๐(5)are negative. Sketch a graph of ๐๐.
I notice that the graph is above the ๐ฅ๐ฅ-axis on either side of the ๐ฅ๐ฅ-intercept at 2. This means that (2, 0) is a relative minimum point for this function.
Because ๐๐(โ2) and ๐๐(0) are positive and ๐๐(โ1) = 0, I know the graph of ๐๐ lies above the ๐ฅ๐ฅ-axis for ๐ฅ๐ฅ < โ1 and โ1 < ๐ฅ๐ฅ < 2 and that it intersects the ๐ฅ๐ฅ-axis at โ1,
Because ๐๐(3) and ๐๐(5) are negative and ๐๐(4) = 0, I know the graph of ๐๐ lies below the ๐ฅ๐ฅ-axis for ๐ฅ๐ฅ > 4 and 2 < ๐ฅ๐ฅ < 4 and that it intersects the ๐ฅ๐ฅ-axis at 4.
Because ๐๐(0) is positive and ๐๐(3) is negative and ๐๐(2) = 0, I know the graph of ๐๐ crosses the ๐ฅ๐ฅ-axis at 2.
For ๐๐ < โ๐๐: Since ๐๐(โ๐๐) = ๐๐๐๐, the graph is above the ๐๐-axis for ๐๐ < โ๐๐.
For โ๐๐ < ๐๐ < ๐๐: Since ๐๐(๐๐) = โ๐๐, the graph is below the๐๐-axis for โ๐๐ < ๐๐ < ๐๐.
For ๐๐ < ๐๐ < ๐๐: Since ๐๐(๐๐.๐๐) = ๐๐.๐๐๐๐๐๐๐๐, the graph is abovethe ๐๐-axis for ๐๐ < ๐๐ < ๐๐.
For ๐๐ > ๐๐: Since ๐๐(๐๐) = ๐๐, the graph is above the ๐๐-axis for๐๐ > ๐๐.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 15: Structure in Graphs of Polynomial Functions
Lesson 15: Structure in Graphs of Polynomial Functions
Characteristics of Polynomial Graphs
1. Graph the functions ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ, ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3, and โ(๐ฅ๐ฅ) = ๐ฅ๐ฅ5 simultaneously using a graphing utility, andzoom in at the origin.
a. At ๐ฅ๐ฅ = 0.8, order the values of the functions from least to greatest.
At ๐๐ = ๐๐.๐๐, ๐๐(๐๐.๐๐) < ๐๐(๐๐.๐๐) < ๐๐(๐๐.๐๐)
b. At ๐ฅ๐ฅ = 2, order the values of the functions from least to greatest.
At ๐๐ = ๐๐, ๐๐(๐๐) < ๐๐(๐๐) < ๐๐(๐๐)
c. At ๐ฅ๐ฅ = 1, order the values of the functions from least to greatest.
At ๐๐ = ๐๐, ๐๐(๐๐) = ๐๐(๐๐) = ๐๐(๐๐)
Since โ is increasing at a faster rate than ๐๐ for ๐ฅ๐ฅ > 1, I can tell that ๐ฅ๐ฅ5 > ๐ฅ๐ฅ3 for ๐ฅ๐ฅ > 1.
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Homework Helper A Story of Functions
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2015-16 M1 ALGEBRA II
Lesson 15: Structure in Graphs of Polynomial Functions
2. The Oak Ridge National Laboratory conducts research for the United States Department of Energy. Thefollowing table contains data from the ORNL about fuel efficiency and driving speed on level roads forvehicles that weigh between 60,000 lb to 70,000 lb.
Fuel Economy Versus Speed for Flat Terrain
Speed (mph)
Fuel Economy (mpg)
51 5.8 53 7.6 55 8.9 57 8.9 59 9.8 61 8.8 63 8.2 65 7.9 67 7.8 69 7.8 71 7.2 73 7.4 75 7.1
Source: www-cta.ornl.gov/cta/Publications/Reports/ORNL TM 2011 471.pdf - 400k - 2011-12-05
a. Plot the data using a graphing utility.
b. Determine if the data display the characteristics of an odd- or even-degree polynomial function.
The characteristics are similar to that of an even-degree polynomial function.
c. List one possible reason the data might have such a shape.
Fuel efficiency decreases at excessive speeds.
The shape of the graph shows that both ends of the graph point downward, so I know that it would be best represented using a polynomial of even degree.
Speed (mph)
Fue
l eco
nom
y (m
pg)
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 15: Structure in Graphs of Polynomial Functions
3. Determine if each of the following functions is even or odd. Explain how you know.a. ๐๐(๐ฅ๐ฅ) = 3๐ฅ๐ฅ3 โ 5๐ฅ๐ฅ
๐๐(โ๐๐) = ๐๐(โ๐๐)๐๐ โ ๐๐(โ๐๐) = โ๐๐๐๐๐๐ + ๐๐๐๐ = โ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐๏ฟฝ = โ๐๐(๐๐)
The function ๐๐ is odd.
b. ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ4 โ 6๐ฅ๐ฅ2 + 4
๐๐(โ๐๐) = ๐๐(โ๐๐)๐๐ โ ๐๐(โ๐๐)๐๐ + ๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐ = ๐๐(๐๐)
The function ๐๐ is even.
c. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 + 1
๐๐(โ๐๐) = (โ๐๐)๐๐ + (โ๐๐)๐๐ + ๐๐ = โ๐๐๐๐ + ๐๐๐๐ + ๐๐
The function ๐๐ is neither even nor odd.
4. Determine the ๐ฆ๐ฆ-intercept for each function below.a. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 โ ๐ฅ๐ฅ + 2
๐๐(๐๐) = ๐๐๐๐ โ ๐๐ + ๐๐ = ๐๐The ๐๐-intercept is ๐๐.
b. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ4 โ ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ
๐๐(๐๐) = ๐๐๐๐ โ ๐๐๐๐ โ ๐๐ = ๐๐The ๐๐-intercept is ๐๐.
I know that ๐๐(โ๐ฅ๐ฅ) = ๐๐(๐ฅ๐ฅ) for even functions and ๐๐(โ๐ฅ๐ฅ) = โ๐๐(๐ฅ๐ฅ) for odd functions. This function does not meet either condition.
I know the ๐ฆ๐ฆ-intercept is the output of a function when the input is 0.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 16: Modeling with PolynomialsโAn Introduction
Lesson 16: Modeling with PolynomialsโAn Introduction
Modeling Real-World Situations with Polynomials
1. For a fundraiser, members of the math club decide to make and sell slices of pie on March 14. They aretrying to decide how many slices of pie to make and sell at a fixed price. They surveyed student interestaround school and made a scatterplot of the number of slices sold (๐ฅ๐ฅ) versus profit in dollars (๐ฆ๐ฆ).
a. Identify the ๐ฅ๐ฅ-intercepts and the ๐ฆ๐ฆ-intercept. Interpret their meaning in context.
The ๐๐-intercepts are approximately ๐๐๐๐ and ๐๐๐๐. The number ๐๐๐๐ represents the number of slices ofpie that would be made and sold for the math club to break even; in other words, the revenuewould be equal to the amount spent on supplies. The number ๐๐๐๐ represents the number of slicesabove which the supply would exceed the demand so that the cost to produce the slices of piewould exceed the revenue.
The ๐๐-intercept is approximately โ๐๐๐๐. The โ๐๐๐๐ represents the money that the math clubmembers must spend on supplies in order to make the pies. That is, they will lose $๐๐๐๐ if they sell ๐๐ slices of pie.
b. How many slices of pie should they sell in order to maximize the profit?
They should sell approximately ๐๐๐๐slices to maximize the profit.
c. What is the maximum profit?
The maximum profit is slightlymore than $๐๐๐๐๐๐.
With a single turning point, I recognize that this data can be modeled using a quadratic polynomial function.
From the graph, I can see that the coordinates of the vertex are approximately (55, 200). The number 55 is the number of slices sold, and the number 200 is the profit, in dollars.
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Homework Helper A Story of Functions
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2015-16 M1 ALGEBRA II
Lesson 16: Modeling with PolynomialsโAn Introduction
2. The following graph shows the average monthly high temperature in degrees Fahrenheit (๐ฅ๐ฅ) in Albany,NY, from May 2013 through April 2015, where ๐ฆ๐ฆ represents the number of months since April 2013.(Source: U.S. Climate Data)
a. What degree polynomial would be a reasonable choice to model this data?
Since the graph has ๐๐ turning points (๐๐relative minima, ๐๐ relative maxima), a degree๐๐ polynomial could be used.
b. Let ๐๐ be the function that represents the temperature, in degrees Fahrenheit, as a function of time๐ฅ๐ฅ, in months. What is the value of ๐๐(10), and what does it represent in the context of the problem?
The value ๐ป๐ป(๐๐๐๐) represents the average monthly high temperature ten months after April 2013,which is February 2014. From the graph, ๐ป๐ป(๐๐๐๐) โ ๐๐๐๐, which means the average monthly hightemperature in February 2014 in Albany was about ๐๐๐๐ยฐ๐ ๐ .
I remember that the maximum number of relative maxima and minima is equal to one less than the degree of the polynomial.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 17: Modeling with PolynomialsโAn Introduction
I see from the graph that the vertical intercept is โ50. I can substitute (0,โ50) into the function and isolate ๐๐ to find its value.
Lesson 17: Modeling with PolynomialsโAn Introduction
Modeling Real-World Situations with Polynomials
1. Recall the math club fundraiser from the Problem Set of the previous lesson. The club members wouldlike to find a function to model their data, so Joe draws a curve through the data points as shown.
a. The function that models the profit in terms of the number of slices of pie made has the form๐๐(๐ฅ๐ฅ) = ๐๐(๐ฅ๐ฅ2 โ 119๐ฅ๐ฅ + 721). Use the vertical intercept labeled on the graph to find the value ofthe leading coefficient ๐๐.
Since ๐ท๐ท(๐๐) = โ๐๐๐๐, we have โ๐๐๐๐ = ๐๐๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐(๐๐) + ๐๐๐๐๐๐๏ฟฝ.
Then โ๐๐๐๐ = ๐๐๐๐๐๐๐๐, so ๐๐ โ โ๐๐.๐๐๐๐.
So ๐ท๐ท(๐๐) = โ๐๐.๐๐๐๐(๐๐๐๐ โ ๐๐๐๐๐๐๐๐+ ๐๐๐๐๐๐) is a reasonable model.
b. From the graph, estimate the profit if the math club sells 50 slices of pie.
Reading from the graph, the profit is approximately $๐๐๐๐๐๐ if the club sells ๐๐๐๐ slices of pie.
c. Use your function to estimate the profit if the math club sells 50 slices of pie.
Because ๐ท๐ท(๐๐๐๐) = โ๐๐.๐๐๐๐๏ฟฝ(๐๐๐๐)๐๐ โ ๐๐๐๐๐๐(๐๐๐๐) + ๐๐๐๐๐๐๏ฟฝ = ๐๐๐๐๐๐.๐๐๐๐, the equation predicts a profit of$๐๐๐๐๐๐.๐๐๐๐.
I notice that the curve used to model the data has one relative maximum and no relative minima, so the curve can be represented with a quadratic function.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 17: Modeling with PolynomialsโAn Introduction
I know the surface area of the container is the sum of the area of the circular base, ๐๐๐๐2, added to the lateral area of the cylinder. I can find the lateral area by multiplying the height of the cylinder by the circumference of the circular base, which is 2๐๐๐๐โ.
I know the volume of a cylinder is the product of the area of its circular base and the height.
I can substitute the expression for โ from part (b) to write an expression for volume in terms of ๐๐.
2. A container is to be constructed as a cylinder with no top.a. Draw and label the sides of the container.
b. The surface area is 150 cm2. Write a formula for the surface area ๐๐, and then solve for โ.
๐บ๐บ = ๐ ๐ ๐๐๐๐ + ๐๐๐ ๐ ๐๐๐ ๐ = ๐๐๐๐๐๐
๐ ๐ =๐๐๐๐๐๐ โ ๐ ๐ ๐๐๐๐
๐๐๐ ๐ ๐๐
c. Write a formula for the function of the volume of the container in terms of ๐๐.
๐ฝ๐ฝ(๐๐) = ๐ ๐ ๐๐๐๐๐ ๐ = ๐ ๐ ๐๐๐๐ ๏ฟฝ๐๐๐๐๐๐ โ ๐ ๐ ๐๐๐๐
๐๐๐ ๐ ๐๐ ๏ฟฝ =๐๐๐๐๐๐๐ ๐ ๐๐๐๐ โ ๐ ๐ ๐๐๐๐๐๐
๐๐๐ ๐ ๐๐= ๐๐๐๐๐๐ โ
๐ ๐ ๐๐๐๐
๐๐
๐๐
๐ ๐
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Homework Helper A Story of Functions
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2015-16 M1 ALGEBRA II
Lesson 17: Modeling with PolynomialsโAn Introduction
I know the ๐ฅ๐ฅ-coordinate of the relative maximum point represents the value of ๐๐ that maximizes volume. I can substitute this value into the expression for part (b) to find โ.
d. Use a graph of the volume as a function of ๐๐ to find the maximum volume of the container.
The maximum volume isapproximately ๐๐๐๐๐๐.๐๐๐๐ ๐๐๐๐๐๐ whenthe radius isapproximately ๐๐.๐๐๐๐ ๐๐๐๐.
e. What dimensions should the container have in order to maximize its volume?
The relative maximum occurs when ๐๐ โ ๐๐.๐๐๐๐. Then ๐ ๐ = ๐๐๐๐๐๐โ๐ ๐ ๐๐๐๐
๐๐๐ ๐ ๐๐โ ๐๐.๐๐๐๐. The container should
have radius approximately ๐๐.๐๐๐๐ ๐๐๐๐ and height approximately ๐๐.๐๐๐๐ ๐๐๐๐ to maximize its volume.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 18: Overcoming a Second Obstacle in FactoringโWhat If There Is a Remainder?
Lesson 18: Overcoming a Second Obstacle in FactoringโWhat If
There Is a Remainder?
Finding a Quotient of Two Polynomials by Inspection
1. For the pair of problems shown, find the first quotient by factoring the numerator. Then, find the secondquotient by using the first quotient.
๐ฅ๐ฅ2โ๐ฅ๐ฅโ12๐ฅ๐ฅโ4
๐๐๐๐โ๐๐โ๐๐๐๐๐๐โ๐๐
= (๐๐โ๐๐)(๐๐+๐๐)(๐๐โ๐๐)
= ๐๐ + ๐๐
๐ฅ๐ฅ2โ๐ฅ๐ฅโ15๐ฅ๐ฅโ4
๐๐๐๐โ๐๐โ๐๐๐๐๐๐โ๐๐
= (๐๐๐๐โ๐๐โ๐๐๐๐)โ๐๐(๐๐โ๐๐)
= (๐๐๐๐โ๐๐โ๐๐๐๐)(๐๐โ๐๐)
โ ๐๐(๐๐โ๐๐)
= ๐๐ + ๐๐ โ ๐๐(๐๐โ๐๐)
2. Rewrite the numerator in the form (๐ฅ๐ฅ โ โ)2 + ๐๐ by completing the square. Then, find the quotient.
๐ฅ๐ฅ2โ2๐ฅ๐ฅโ1๐ฅ๐ฅโ1
Once I factor the numerator, I can see that the numerator and denominator have a common factor by which I can divide, if we assume that ๐ฅ๐ฅ โ 4 โ 0.
To complete the square with the expression ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ, I need to add a constant that is the square of half the linear coefficient, which means I need to add 1. To maintain the value of the numerator ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ โ 1, I can rewrite it as (๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ + 1) โ 1 โ 1, and then I can rewrite the expression in the form (๐ฅ๐ฅ โ โ)2 + ๐๐.
I can see that the numerator of this rational expression can be rewritten as the difference of the numerator of the first expression and 3, which means the quotient will be the same as in the first expression, but I will have a remainder of โ3.
๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐
=๏ฟฝ๐๐๐๐ โ ๐๐๐๐๏ฟฝ โ ๐๐
๐๐ โ ๐๐
=๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๏ฟฝ โ ๐๐ โ ๐๐
๐๐ โ ๐๐
=(๐๐๐๐ โ ๐๐๐๐ + ๐๐) โ ๐๐
๐๐ โ ๐๐
=๐๐๐๐ โ ๐๐๐๐ + ๐๐
๐๐ โ ๐๐โ
๐๐๐๐ โ ๐๐
=(๐๐ โ ๐๐)๐๐
๐๐ โ ๐๐โ
๐๐๐๐ โ ๐๐
= (๐๐ โ ๐๐) โ๐๐
๐๐ โ ๐๐
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 18: Overcoming a Second Obstacle in FactoringโWhat If There Is a Remainder?
Finding a Quotient of Two Polynomials by the Reverse Tabular Method
3. Find the quotient by using the reverse tabular method.
๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ + 1๐ฅ๐ฅ โ 2
๐๐๐๐ ๐๐๐๐ โ๐๐
๐๐๐๐ ๐๐๐๐๐๐ โ๐๐๐๐ ๐๐
๐๐๐๐ โ๐๐๐๐๐๐ โ๐๐๐๐ ๐๐ โ๐๐
๐๐๐๐๐๐ โ๐๐๐๐ ๐๐ = ๐๐ โ ๐๐
๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ โ ๐๐
=๐๐๐๐ โ ๐๐๐๐ + ๐๐ โ ๐๐
๐๐ โ ๐๐=๐๐๐๐ โ ๐๐๐๐ + ๐๐
๐๐ โ ๐๐โ
๐๐๐๐ โ ๐๐
= ๐๐๐๐ + ๐๐๐๐ โ ๐๐ โ๐๐
๐๐ โ ๐๐
Finding a Quotient of Two Polynomials Using Long Division
4. Find the quotient by using long division.
๐๐๐๐ โ๐๐๐๐๐๐ +๐๐๐๐ +๐๐ โ๐๐
๐๐+ ๐๐
๐๐ + ๐๐ ๐๐๐๐ โ๐๐๐๐๐๐ +๐๐๐๐๐๐ +๐๐๐๐ +๐๐
โ(๐๐๐๐ +๐๐๐๐)
โ๐๐๐๐๐๐ +๐๐๐๐๐๐
โ(โ๐๐๐๐๐๐ โ๐๐๐๐๐๐)
๐๐๐๐๐๐ +๐๐๐๐
โ(๐๐๐๐๐๐ +๐๐๐๐)
๐๐๐๐ +๐๐
โ(๐๐๐๐ +๐๐)
โ๐๐
The remainder is โ๐๐.
This is the method I used to divide polynomials in Lesson 3.
The remainder is equal to the difference between the constant in the dividend and the constant in the table.
This is the method I used to divide polynomials in Lesson 5.
Using placeholders helps me to align like terms as I carry out the long division procedure.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 19: The Reminder Theorem
The remainder theorem tells me that when polynomial ๐๐ is divided by ๏ฟฝ๐ฅ๐ฅ โ (โ3)๏ฟฝ, the remainder is the value of ๐๐ evaluated at โ3.
The remainder theorem tells me that ๐๐(3) has the same value as the remainder to the division of ๐๐ by (๐ฅ๐ฅ โ 3).
Since the remainder theorem states that ๐๐ divided by (๐ฅ๐ฅ โ 1) has a remainder equal to ๐๐(1), I know ๐๐(1) = 10.
Lesson 19: The Remainder Theorem
Applying the Remainder Theorem to Find Remainders and Evaluate Polynomials at Specific Inputs
1. Use the remainder theorem to find the remainder for the following division.
๐ฅ๐ฅ4 + 2๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ + 1๐ฅ๐ฅ + 3
Let ๐ท๐ท(๐๐) = ๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐. Then
๐ท๐ท(โ๐๐) = (โ๐๐)๐๐ + ๐๐(โ๐๐)๐๐ โ ๐๐(โ๐๐) + ๐๐ = ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ + ๐๐ = ๐๐๐๐.
Therefore, the remainder of the division ๐๐๐๐+๐๐๐๐๐๐โ๐๐๐๐+๐๐
๐๐+๐๐ is ๐๐๐๐.
2. Consider the polynomial ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + 2. Find ๐๐(3) in two ways.
๐ท๐ท(๐๐) = (๐๐)๐๐ + (๐๐)๐๐ โ ๐๐(๐๐) + ๐๐ = ๐๐๐๐
๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ โ ๐๐
= ๐๐๐๐ + ๐๐๐๐ + ๐๐ +๐๐๐๐๐๐ โ ๐๐
so ๐ท๐ท(๐๐) = ๐๐๐๐.
3. Find the value ๐๐ so that ๐๐๐๐๐๐โ๐๐+๐๐๐๐โ๐๐
has remainder 10.
Let ๐ท๐ท(๐๐) = ๐๐๐๐๐๐ โ ๐๐ + ๐๐. Then ๐ท๐ท(๐๐) = ๐๐๐๐, so
๐๐๏ฟฝ๐๐๐๐๏ฟฝ โ ๐๐ + ๐๐ = ๐๐๐๐๐๐๐๐ โ ๐๐ = ๐๐๐๐
๐๐ =๐๐๐๐๐๐
.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 19: The Reminder Theorem
By the factor theorem, I know that if ๐๐(๐๐) = 0, then (๐ฅ๐ฅ โ ๐๐) is a factor of ๐๐. By the remainder theorem, I know that if (๐ฅ๐ฅ โ ๐๐) is a factor of ๐๐, then ๐๐(๐๐) = 0. This means that if ๐๐(๐๐) โ 0, then (๐ฅ๐ฅ โ ๐๐) is not a factor of ๐๐.
I can find this quotient using long division or the reverse tabular method. I can then factor this quotient as the product of two linear factors.
I can substitute inputs into ๐๐ whose values are between the values of the zeros to help discover the shape of the graph. For example, ๐๐(0) =โ3, so the graph is beneath the ๐ฅ๐ฅ-axis between the ๐ฅ๐ฅ-values of โ1 and 1.
Applying the Factor Theorem
4. Determine whether the following are factors of the polynomial ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 โ 4๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 6.a. (๐ฅ๐ฅ โ 2)
๐ท๐ท(๐๐) = (๐๐)๐๐ โ ๐๐(๐๐)๐๐ + (๐๐) + ๐๐ = ๐๐
Yes, ๐๐ โ ๐๐ is a factor of ๐ท๐ท.
b. (๐ฅ๐ฅ โ 1)
๐ท๐ท(๐๐) = (๐๐)๐๐ โ ๐๐(๐๐)๐๐ + (๐๐) + ๐๐ = ๐๐
No, ๐๐ โ ๐๐ is not a factor of ๐ท๐ท.
Using the Remainder Theorem to Graph Polynomials
5. Consider the polynomial function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + 3๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 3.a. Verify that ๐๐(โ1) = 0.
๐ท๐ท(โ๐๐) = (โ๐๐)๐๐ + ๐๐(โ๐๐)๐๐ โ (โ๐๐) โ ๐๐ = ๐๐
b. Since ๐๐(โ1) = 0, what are the factors of polynomial ๐๐?
Since ๐ท๐ท(โ๐๐) = ๐๐, (๐๐ + ๐๐) is a factor of ๐ท๐ท.
Since ๐๐๐๐+๐๐๐๐๐๐โ๐๐โ๐๐
๐๐+๐๐= ๐๐๐๐ + ๐๐๐๐ โ ๐๐, it follows that
๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐ โ ๐๐ = (๐๐ + ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ โ ๐๐๏ฟฝ = (๐๐ + ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐).
The factors of ๐ท๐ท are ๐๐ + ๐๐,๐๐ + ๐๐ and ๐๐ โ ๐๐.
c. Find the zeros of ๐๐, and then use the zeros to sketch the graph of ๐๐.
The zeros of ๐ท๐ท are โ๐๐, โ ๐๐, and ๐๐.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 19: The Reminder Theorem
Using the Factor Theorem to Write Equations for Polynomials
6. A fourth-degree polynomial function ๐๐ is graphed at right.
a. Write a formula for ๐๐ in factored form using ๐๐ for theconstant factor.
๐๐(๐๐) = ๐๐(๐๐ + ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ โ ๐๐)
b. Use the fact that ๐๐(โ4) = โ120 to find the constant factor ๐๐.โ๐๐๐๐๐๐ = ๐๐(โ๐๐+ ๐๐)(โ๐๐ + ๐๐)(โ๐๐โ ๐๐)(โ๐๐ โ ๐๐)โ๐๐๐๐๐๐ = โ๐๐๐๐๐๐๐๐
๐๐ = ๐๐ ๐๐(๐๐) = (๐๐ + ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ โ ๐๐)
Since ๐๐ has zeros at โ6,โ2, 1, and 2, I know ๐๐ has factors (๐ฅ๐ฅ + 6), (๐ฅ๐ฅ + 2), (๐ฅ๐ฅ โ 1), and (๐ฅ๐ฅ โ 2). I know that since ๐๐ is a fourth-degree polynomial with four zeros, each zero has a multiplicity of 1.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 20: Modeling Riverbeds with Polynomials
Lesson 20: Modeling Riverbeds with Polynomials
This lesson requires students to fit a polynomial function to data values in order to model a riverbed using a polynomial function.
1. Use the remainder theorem to find the polynomial ๐๐ of least degree so that ๐๐(3) = 1 and ๐๐(5) = โ3.
๐ท๐ท(๐๐) = (๐๐ โ ๐๐)๐๐(๐๐) + ๐๐
๐ท๐ท(๐๐) = (๐๐ โ ๐๐)๐๐(๐๐) โ ๐๐
(๐๐ โ ๐๐)๐๐(๐๐) + ๐๐ = (๐๐ โ ๐๐)๐๐(๐๐) โ ๐๐ ๐๐๐๐(๐๐) โ ๐๐๐๐(๐๐) + ๐๐ = ๐๐๐๐(๐๐) โ ๐๐๐๐(๐๐)โ ๐๐
๐๐๐๐(๐๐) = โ๐๐ ๐๐(๐๐) = โ๐๐
๐ท๐ท(๐๐) = (๐๐ โ ๐๐)(โ๐๐) + ๐๐ = โ๐๐๐๐ + ๐๐
Check: ๐ท๐ท(๐๐) = โ๐๐(๐๐) + ๐๐ = ๐๐ ๐ท๐ท(๐๐) = โ๐๐(๐๐) + ๐๐ = โ๐๐.
2. Write a quadratic function ๐๐ such that ๐๐(0) = 4, ๐๐(2) = โ4, and ๐๐(5) = โ1 using a system ofequations.
๐ท๐ท(๐๐) = ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐
๐๐ = ๐๐
๐ท๐ท(๐๐) = ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐
๐ท๐ท(๐๐) = ๐๐(๐๐)๐๐ + ๐๐(๐๐) + ๐๐ = โ๐๐, so ๐๐๐๐ + ๐๐๐๐ = โ๐๐
๐ท๐ท(๐๐) = ๐๐(๐๐)๐๐ + ๐๐(๐๐) + ๐๐ = โ๐๐, so ๐๐๐๐๐๐ + ๐๐๐๐ = โ๐๐
๐๐(๐๐๐๐ + ๐๐๐๐) = ๐๐(โ๐๐) so ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = โ๐๐๐๐ โ๐๐(๐๐๐๐๐๐+ ๐๐๐๐) = โ๐๐(โ๐๐) so โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐
Adding the two equations gives โ๐๐๐๐๐๐ = โ๐๐๐๐, so ๐๐ = ๐๐.
๐๐(๐๐) + ๐๐๐๐ = โ๐๐, so ๐๐ = โ๐๐.
๐ท๐ท(๐๐) = ๐๐๐๐ โ ๐๐๐๐ + ๐๐
The remainder theorem tells me that for any number ๐๐, I can find some polynomial ๐๐ so that ๐๐ = (๐ฅ๐ฅ โ ๐๐)๐๐(๐ฅ๐ฅ) + ๐๐(๐๐).
I can evaluate ๐๐ at 2 and at 5 and set the expressions that result equal to the known outputs to create a system of two linear equations. I can solve the system using elimination.
Once I found the value of ๐๐, I back-substituted its value into the first linear equation to find the value of ๐๐.
Since ๐๐(0) = 4 and the ๐ฆ๐ฆ-intercept of ๐๐ is ๐๐, I know that ๐๐ = 4.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 20: Modeling Riverbeds with Polynomials
Representing Data with a Polynomial Using the Remainder Theorem
To use this strategy, remember that ๐๐(๐ฅ๐ฅ) = (๐ฅ๐ฅ โ ๐๐)๐๐(๐ฅ๐ฅ) + ๐๐, where ๐๐ is a polynomial with degree one less than ๐๐ and ๐๐ is the remainder. The remainder theorem tells us that ๐๐ = ๐๐(๐๐).
3. Find a degree-three polynomial function ๐๐ such that ๐๐(โ2) = 0, ๐๐(0) = 4, ๐๐(2) = 16, and ๐๐(3) = 55.Use the table below to organize your work. Write your answer in standard form, and verify by showingthat each point satisfies the equation.
Function Value
Substitute the data point into the current form of the equation for ๐ท๐ท.
Apply the remainder
theorem to ๐๐, ๐๐, or ๐๐.
Rewrite the equation for ๐ท๐ท in terms of ๐๐, ๐๐, or ๐๐.
๐ท๐ท(โ๐๐) = ๐๐
๐ท๐ท(๐๐) = (๐๐ + ๐๐)๐๐(๐๐)
๐ท๐ท(๐๐) = ๐๐
๐ท๐ท(๐๐) = (๐๐ + ๐๐)๐๐(๐๐) ๐ท๐ท(๐๐) = (๐๐ + ๐๐)๐๐(๐๐)
๐๐ = ๐๐๐๐(๐๐) ๐๐(๐๐) = ๐๐
๐๐(๐๐) = ๐๐๐๐(๐๐) + ๐๐ ๐ท๐ท(๐๐) = (๐๐ + ๐๐)(๐๐๐๐(๐๐) + ๐๐)
๐ท๐ท(๐๐) = ๐๐๐๐
๐ท๐ท(๐๐) = (๐๐ + ๐๐)(๐๐๐๐(๐๐) + ๐๐) ๐ท๐ท(๐๐) = (๐๐ + ๐๐)(๐๐๐๐(๐๐) + ๐๐) ๐๐๐๐ = ๐๐(๐๐๐๐(๐๐) + ๐๐) ๐๐ = ๐๐๐๐(๐๐) + ๐๐
๐๐(๐๐) = ๐๐
๐๐(๐๐) = (๐๐ โ ๐๐)๐๐(๐๐)+ ๐๐ ๐ท๐ท(๐๐) = (๐๐ + ๐๐)[๐๐[(๐๐ โ ๐๐)๐๐(๐๐) + ๐๐] + ๐๐]
๐ท๐ท(๐๐) = ๐๐๐๐
๐ท๐ท(๐๐) = (๐๐ + ๐๐)[๐๐[(๐๐ โ ๐๐)๐๐(๐๐) + ๐๐] + ๐๐] ๐ท๐ท(๐๐) = (๐๐ + ๐๐)[๐๐(๐๐ โ ๐๐)๐๐(๐๐) + ๐๐] + ๐๐] ๐๐๐๐ = ๐๐[๐๐(๐๐(๐๐) + ๐๐) + ๐๐] ๐๐๐๐ = ๐๐๐๐(๐๐) + ๐๐
๐๐(๐๐) = ๐๐
๐๐(๐๐) = ๐๐ ๐ท๐ท(๐๐) = (๐๐ + ๐๐)[๐๐[(๐๐ โ ๐๐)(๐๐) + ๐๐] + ๐๐]
Then, ๐ท๐ท(๐๐) = (๐๐ + ๐๐)[๐๐[(๐๐ โ ๐๐)(๐๐) + ๐๐] + ๐๐] = (๐๐ + ๐๐)[๐๐(๐๐๐๐ โ ๐๐) + ๐๐] = (๐๐ + ๐๐)(๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐).
Expanding and combining like terms gives the polynomial ๐ท๐ท(๐๐) = ๐๐๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐.
๐ท๐ท(โ๐๐) = ๐๐(โ๐๐)๐๐ + (โ๐๐)๐๐ โ ๐๐(โ๐๐) + ๐๐ = โ๐๐๐๐+ ๐๐ + ๐๐ + ๐๐ = ๐๐ ๐ท๐ท(๐๐) = ๐๐(๐๐)๐๐ + (๐๐)๐๐ โ ๐๐(๐๐) + ๐๐ = ๐๐ ๐ท๐ท(๐๐) = ๐๐(๐๐)๐๐ + (๐๐)๐๐ โ ๐๐(๐๐) + ๐๐ = ๐๐๐๐ + ๐๐ โ ๐๐ + ๐๐ = ๐๐๐๐ ๐ท๐ท(๐๐) = ๐๐(๐๐)๐๐ + (๐๐)๐๐ โ ๐๐(๐๐) + ๐๐ = ๐๐๐๐ + ๐๐ โ ๐๐๐๐ + ๐๐ = ๐๐๐๐
I can replace ๐๐(๐ฅ๐ฅ) with the equivalent expression ๐ฅ๐ฅ๐๐(๐ฅ๐ฅ) + 2.
Since ๐๐(โ2) = 0, I know (๐ฅ๐ฅ + 2) is a factor of ๐๐. Then there is some polynomial ๐๐ so that ๐๐(๐ฅ๐ฅ) = (๐ฅ๐ฅ + 2)๐๐(๐ฅ๐ฅ).
The remainder theorem tells me ๐๐(๐ฅ๐ฅ) = (๐ฅ๐ฅ โ 0)๐๐(๐ฅ๐ฅ) + ๐๐(0).
I can evaluate the function at the given inputs to check my answer.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 21: Modeling Riverbeds with Polynomials
Lesson 21: Modeling Riverbeds with Polynomials
This lesson addresses modeling a cross-section of a riverbed with a polynomial function using technology. Measurements of depth are provided along a riverbed, and these values are used to create a geometric model for the cross-sectional area of the riverbed. This model in turn is used to compute the volumetric flow of the river.
1. Suppose that depths of the riverbed were measured for a cross-section of a different river. Assume thecross-sectional area of the river can be modeled with polynomial ๐๐.a. Based on the depths provided, sketch the cross-section of the river, and estimate its area.
๐๐(0) = 0 ๐๐(25) = โ20.5 ๐๐(60) = โ25 ๐๐(95) = โ32 ๐๐(132.5) = โ17 ๐๐(165) = โ22.5 ๐๐(200) = 0
Area estimates:
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐)(๐๐๐๐.๐๐) = ๐๐๐๐๐๐.๐๐๐๐
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐.๐๐ + ๐๐๐๐)(๐๐๐๐) = ๐๐๐๐๐๐.๐๐๐๐
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐)(๐๐๐๐) = ๐๐๐๐๐๐.๐๐
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐.๐๐)(๐๐๐๐.๐๐) = ๐๐๐๐๐๐๐๐.๐๐๐๐๐๐
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐.๐๐)(๐๐๐๐.๐๐) = ๐๐๐๐๐๐.๐๐๐๐๐๐
๐จ๐จ๐๐ =๐๐๐๐
(๐๐๐๐)(๐๐๐๐.๐๐) = ๐๐๐๐๐๐
So, the total area can be estimated by
๐จ๐จ = ๐จ๐จ๐๐ + ๐จ๐จ๐๐ + ๐จ๐จ๐๐ + ๐จ๐จ๐๐ + ๐จ๐จ๐๐ + ๐จ๐จ๐๐ = ๐๐๐๐๐๐๐๐.
The cross-sectional area is approximately ๐๐๐๐๐๐๐๐ square feet.
I can estimate the area by dividing the riverbed cross-section into triangular and trapezoidal sections, finding the area of each section, and finding the sum of these areas.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 21: Modeling Riverbeds with Polynomials
b. Suppose that the speed of the water was measured at 200 ftmin
. What is the approximate volumetric flow in this section of the river, measured in gallons per minute?
The volumetric flow is approximately ๐๐๐๐๐๐๐๐ ๐๐๐ญ๐ญ๐๐ ๏ฟฝ๐๐๐๐๐๐ ๐๐๐ญ๐ญ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ
๏ฟฝ โ ๐๐๐๐๐๐โ๐๐๐๐๐๐ ๐๐๐ญ๐ญ๐๐
๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ. Converting to gallons per
minute, this is
๏ฟฝ๐๐๐๐๐๐โ๐๐๐๐๐๐ ๐๐๐ญ๐ญ๐๐
๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๏ฟฝ๏ฟฝ๐๐.๐๐๐๐๐๐๐๐๐๐
๐ ๐ ๐ ๐ ๐ ๐ ๐๐๐ญ๐ญ๐๐
๏ฟฝ โ ๐๐โ๐๐๐๐๐๐โ๐๐๐๐๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ
.
I know 1 ft3 is equivalent to 7.48052 gallons, and I can use this equivalency to convert the flow
rate from ft3
min to gallons per minute.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 22: Equivalent Rational Expressions
Lesson 22: Equivalent Rational Expressions
This lesson addresses generating equivalent rational expressions by multiplying or dividing the numerator and denominator by the same factor and noting restrictions for values of variables.
Reduce Rational Expressions to Lowest Terms
1. Find an equivalent rational expression in lowest terms, and identify the value(s) of the variable that mustbe excluded to prevent division by zero.
a. 3๐๐โ12๐๐2
18๐๐๐๐๐๐(๐๐โ๐๐๐๐)
๐๐๐๐โ๐๐= ๐๐โ๐๐๐๐
๐๐ where ๐๐ โ ๐๐
b. ๐๐๐๐+๐๐๐๐๐๐๐๐
๐๐(๐๐+๐๐)๐ ๐ ๐๐
= ๐๐+๐๐๐ ๐
where ๐๐ โ ๐๐ and ๐ ๐ โ ๐๐.
c. 2๐๐2+๐๐โ103๐๐โ6
(๐๐๐๐+๐๐)(๐๐โ๐๐)๐๐(๐๐โ๐๐)
= ๐๐๐๐+๐๐๐๐
where ๐๐ โ ๐๐
d. (๐ฅ๐ฅโ๐ฆ๐ฆ)2โ4๐๐2
3๐ฆ๐ฆโ3๐ฅ๐ฅโ6๐๐
2. Write a rational expression with denominator 6๐๐ that is equivalent to 23.
3. Simplify the following rational expression without using a calculator: ๐๐๐๐โ๐๐๐๐๐๐โ10๐๐๐๐๐๐โ๐๐๐๐
.
๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐ โ ๐๐๐๐
=(๐๐๐๐)๐๐ โ (๐๐ โ ๐๐๐๐)๐๐ โ ๐๐ โ ๐๐
(๐๐๐๐ โ ๐๐)๐๐ โ ๐๐ โ ๐๐ โ ๐๐=๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐ โ ๐๐ โ ๐๐
=๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐
= ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐
I know ๐๐ = 0 is a restricted value because it makes the denominator of the original rational expression zero.
I can split the linear term or use the reverse tabular method to factor the trinomial 2๐๐2 + ๐๐ โ 10 into the product of two linear factors.
I recognize the numerator as a difference of two squares.
I need to multiply the denominator of 3 by 2๐๐ to make the denominator 6๐๐.
๐๐๐๐๏ฟฝ๐๐๐๐๐๐๐๐๏ฟฝ = ๐๐๐๐
๐๐๐๐
๏ฟฝ(๐๐โ๐๐)โ๐๐๐๐๏ฟฝ๏ฟฝ(๐๐โ๐๐)+๐๐๐๐๏ฟฝโ๐๐(๐๐โ๐๐+๐๐๐๐)
= ๐๐โ๐๐โ๐๐๐๐โ๐๐
where ๐๐ โ ๐๐โ๐๐๐๐
I can also write this statement as ๐ฅ๐ฅ โ ๐ฆ๐ฆ + 2๐๐ or ๐ฆ๐ฆ โ ๐ฅ๐ฅ โ 2๐๐.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 23: Comparing Rational Functions
Lesson 23: Comparing Rational Expressions
Writing Rational Expressions in Equivalent Forms
1. Rewrite each rational expression as an equivalent rational expression so that all expressions have acommon denominator.๐๐
๐๐๐๐โ2๐๐, ๐๐๐๐๐๐
, ๐๐๐๐+๐๐๐๐๐๐โ๐๐
๐๐๐๐(๐๐ โ ๐๐)
,๐๐๐๐๐๐
,๐๐๐๐ + ๐๐
(๐๐ โ ๐๐)(๐๐ + ๐๐)
The least common denominator is ๐๐๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐).
๐๐(๐๐)(๐๐ + ๐๐)๐๐๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐)
,๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐)๐๐๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐)
,(๐๐๐๐ + ๐๐)(๐๐๐๐)
(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐๐๐)
Analyze Denominators of Rational Expressions
2. For positive ๐ฅ๐ฅ, determine when the following rational expressions have negative denominators.
a. ๐๐+๐๐โ๐๐๐๐+๐๐๐๐โ18
For any real number ๐๐, โ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ is always negative. โ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ = โ๐๐๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐๏ฟฝ โ ๐๐๐๐ + ๐๐๐๐ = โ๐๐(๐๐ โ ๐๐)๐๐ โ ๐๐,
which is the sum of a nonpositive number and a negative number.
b. ๐๐๐๐๐๐
(๐๐โ๐๐)(๐๐+๐๐)(๐ฅ๐ฅ+5)
For positive ๐๐, ๐๐ + ๐๐ and ๐๐ + ๐๐ are always positive. The number ๐๐ โ ๐๐ is negative when ๐๐ < ๐๐, so the denominator is negative when ๐๐ < ๐๐ < ๐๐.
Factoring the denominators will help me determine which factors I need to include in the least common denominator.
I need to make sure to multiply the numerator and denominator of a fraction by the same factor(s) so the value stays the same.
Since this expression is the product of โ1 and a squared number (which cannot be negative), I know this expression cannot be positive.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 23: Comparing Rational Functions
Comparing Rational Expressions
3. If ๐ฅ๐ฅ is a positive number, for which values of ๐ฅ๐ฅ is ๐ฅ๐ฅ > 1๐ฅ๐ฅ?
Since ๐๐ is a positive number, ๐๐ = ๐๐๐๐
(๐๐) = ๐๐๐๐
๐๐.
If ๐๐๐๐
๐๐> ๐๐
๐๐, then ๐๐๐๐ > ๐๐, so ๐๐ > ๐๐.
4. Determine whether the comparison ๐ฅ๐ฅ๐ฅ๐ฅโ1
< ๐ฅ๐ฅ๐ฅ๐ฅโ5
is true for all positive values of ๐ฅ๐ฅ, given that 66โ1
< 66โ5
.
Assuming ๐๐ โ ๐๐ and ๐๐ โ ๐๐, ๐๐๐๐โ๐๐
= ๐๐(๐๐โ๐๐)(๐๐โ๐๐)(๐๐โ๐๐)
and ๐๐๐๐โ๐๐
= ๐๐(๐๐โ๐๐)(๐๐โ๐๐)(๐๐โ๐๐)
.
If ๐๐๐๐โ๐๐
< ๐๐๐๐โ๐๐
, then ๐๐(๐๐โ๐๐)(๐๐โ๐๐)(๐๐โ๐๐)
< ๐๐(๐๐โ๐๐)(๐๐โ๐๐)(๐๐โ๐๐)
and ๐๐๐๐ โ ๐๐๐๐ < ๐๐๐๐ โ ๐๐ whenever (๐๐ โ ๐๐)(๐๐ โ ๐๐) > ๐๐.
However, if (๐๐ โ ๐๐)(๐๐ โ ๐๐) < ๐๐, then ๐๐๐๐ โ ๐๐๐๐ > ๐๐๐๐ โ ๐๐, which is only true for ๐๐ < ๐๐. For ๐๐ < ๐๐ < ๐๐,๐๐
๐๐โ๐๐> ๐๐
๐๐โ๐๐.
5. A large class has ๐ด๐ด students, and a smaller class has ๐ต๐ต students. Use this information to compare thefollowing expressions:
a. ๐ด๐ด๐ด๐ด+๐ต๐ต
and ๐ด๐ด๐ต๐ต
Since ๐จ๐จ is positive, the number ๐จ๐จ + ๐ฉ๐ฉ > ๐ฉ๐ฉ, which means ๐๐๐จ๐จ+๐ฉ๐ฉ
< ๐๐๐ฉ๐ฉ
and ๐จ๐จ๐จ๐จ+๐ฉ๐ฉ
< ๐จ๐จ๐ฉ๐ฉ
.
The expression ๐จ๐จ๐จ๐จ+๐ฉ๐ฉ
represents the fraction of the total students in class ๐จ๐จ, which should be greater
than ๐๐๐๐
and less than ๐๐. The expression ๐จ๐จ๐ฉ๐ฉ
is the ratio of the students in class ๐จ๐จ to the students in
class ๐ฉ๐ฉ, which is greater than ๐๐. Therefore, ๐จ๐จ๐จ๐จ+๐ฉ๐ฉ
< ๐จ๐จ๐ฉ๐ฉ
.
If the denominator is negative, then multiplying the inequality by the denominator will change the direction of the inequality symbol.
I could use a counterexample such as ๐ฅ๐ฅ = 2 to show the inequality is not true for every positive value of ๐ฅ๐ฅ.
The inequality is only true if ๐ฅ๐ฅ > 1 or ๐ฅ๐ฅ <โ1, but the problem states ๐ฅ๐ฅ is positive.
I know this because ๐ด๐ด > ๐ต๐ต.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
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Lesson 23: Comparing Rational Functions
b. ๐ต๐ต๐ด๐ด+๐ต๐ต
and 12
Since ๐จ๐จ > ๐ฉ๐ฉ, adding ๐ฉ๐ฉ ๐๐o both sides gives ๐จ๐จ + ๐ฉ๐ฉ > ๐ฉ๐ฉ +๐ฉ๐ฉ. It follows that ๐๐
๐จ๐จ+๐ฉ๐ฉ< ๐๐
๐ฉ๐ฉ+๐ฉ๐ฉ.
Since ๐ฉ๐ฉ > ๐๐, multiplying both sides of the inequality by ๐ฉ๐ฉ does not change the direction of the inequality, so ๐ฉ๐ฉ
๐จ๐จ+๐ฉ๐ฉ< ๐ฉ๐ฉ
๐ฉ๐ฉ+๐ฉ๐ฉ.
Since ๐ฉ๐ฉ๐ฉ๐ฉ+๐ฉ๐ฉ
= ๐ฉ๐ฉ๐๐๐ฉ๐ฉ
= ๐๐๐๐, it follows that ๐ฉ๐ฉ
๐จ๐จ+๐ฉ๐ฉ< ๐๐
๐๐. This means
that less than half of the total students are in class ๐ฉ๐ฉ.
The expression ๐ต๐ต๐ด๐ด+๐ต๐ต
represents the fraction of total students in class ๐ต๐ต.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 24: Multiplying and Dividing Rational Expressions
Lesson 24: Multiplying and Dividing Rational Expressions
Multiply or Divide Rational Expressions
1. Complete the following operation.
a. 95๏ฟฝ๐ฅ๐ฅ+ 1
3๏ฟฝ รท 3
10
๐๐๐๐๏ฟฝ๐๐ +
๐๐๐๐๏ฟฝ รท
๐๐๐๐๐๐
=๐๐๐๐๏ฟฝ๐๐ +
๐๐๐๐๏ฟฝ โ๐๐๐๐๐๐
=๐๐๐๐๐๐๐๐
๏ฟฝ๐๐ +๐๐๐๐๏ฟฝ = ๐๐๏ฟฝ๐๐ +
๐๐๐๐๏ฟฝ = ๐๐๐๐ + ๐๐
2. Write each rational expression as an equivalent rational expression in lowest terms.
a. ๐๐๐๐๐๐+๐๐๐๐โ๐๐๐๐๐๐๐๐โ๐๐๐๐
โ ๐๐๐๐๐๐โ๐๐๐๐๐๐โ๐๐๐๐
โ๐๐๐๐๐๐โ๐ฅ๐ฅ+2
(๐๐๐๐ โ ๐๐)(๐๐ + ๐๐)๐๐(๐๐ โ ๐๐)(๐๐ + ๐๐)
โ(๐๐ โ ๐๐)(๐๐๐๐ + ๐๐)
โ๐๐(๐๐๐๐ โ ๐๐)(๐๐๐๐ + ๐๐) = โ๐๐๐๐
b. ๏ฟฝ ๐๐โ๐๐๐๐๐๐+๐๐
๏ฟฝโ๐๐
รท ๏ฟฝ๐๐๐๐โ๐๐๐๐+๐๐๐๐๐๐+๐๐๐๐โ๐๐
๏ฟฝ
๏ฟฝ๐๐๐๐ + ๐๐๐๐ โ ๐๐ ๏ฟฝ
๐๐
รท(๐๐ โ ๐๐)๐๐
(๐๐ โ ๐๐)(๐๐ + ๐๐)=
(๐๐๐๐ + ๐๐)๐๐
(๐๐ โ ๐๐)๐๐โ
(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)๐๐
=(๐๐๐๐ + ๐๐)๐๐(๐๐ + ๐๐)
(๐๐ โ ๐๐)๐๐
c. ๏ฟฝ๐๐๐๐+๐๐๐๐โ๐๐๐๐๐๐+๐๐๐๐โ๐๐
๏ฟฝ
๏ฟฝ๐๐๐๐+๐๐๐๐โ๐๐๐๐๐๐+๐๐ ๏ฟฝ
๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ โ ๐๐
รท๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐
๐๐ + ๐๐=
(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)
โ๐๐ + ๐๐
(๐๐ + ๐๐)(๐๐ โ ๐๐) =๐๐
๐๐ โ ๐๐
I recognize that 3๐ฅ๐ฅ2 โ 75 is a difference of squares multiplied by the constant 3. I can factor the remaining expressions the way we did earlier in the module.
I see the only common factor in both the numerator and denominator of the product is ๐ฅ๐ฅ โ 1.
I know this complex fraction can be written as the quotient of the numerator divided by the denominator.
I remember that division is multiplication by the reciprocal.
ยฉ 2015 Great Minds eureka-math.org
49
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 24: Multiplying and Dividing Rational Expressions
3. Determine which of the following numbers is larger without using a calculator, 1215
1512 or 18
20
2018.
๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐รท๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐=๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐โ๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐=๏ฟฝ๐๐๐๐๏ฟฝ
๐๐๐๐โ ๐๐
๐๐๐๐โ (๐๐๐๐)๐๐๐๐ โ ๐๐๐๐๐๐
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐๐
=๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐
๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐=๐๐๐๐๐๐ โ ๐๐๐๐
๐๐๐๐๐๐
= ๏ฟฝ๐๐๐๐๐๐๏ฟฝ๐๐
โ ๐๐๐๐๐๐
๐๐๐๐๐๐= ๏ฟฝ
๐๐๐๐๐๐๏ฟฝ๐๐
โ ๏ฟฝ๐๐๐๐
๐๐๐๐๏ฟฝ๐๐
= ๏ฟฝ๐๐๐๐๐๐๏ฟฝ๐๐
โ ๏ฟฝ๐๐๐๐๐๐๐๐๐๐๐๐
๏ฟฝ๐๐
Since ๐๐๐๐๐๐
> ๐๐, and ๐๐๐๐๐๐๐๐๐๐๐๐
> ๐๐, the product ๏ฟฝ๐๐๐๐๐๐๏ฟฝ๐๐โ ๏ฟฝ๐๐๐๐๐๐
๐๐๐๐๐๐๏ฟฝ๐๐
> ๐๐.
This means that ๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐> ๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐.
4. One of two numbers can be represented by the rational expression ๐๐๐๐โ๐๐
, where ๐ฅ๐ฅ โ 0 and ๐ฅ๐ฅ โ 4.
Find a representation of the second number if the product of the two numbers is 2.
๏ฟฝ๐๐
๐๐ โ ๐๐๏ฟฝ๐๐ = ๐๐
๏ฟฝ ๐๐๐๐ โ ๐๐๏ฟฝ๐๐
๏ฟฝ ๐๐๐๐ โ ๐๐๏ฟฝ
=๐๐
๏ฟฝ ๐๐๐๐ โ ๐๐๏ฟฝ
๐๐ =๐๐
๏ฟฝ ๐๐๐๐ โ ๐๐๏ฟฝ
= ๐๐ โ๐๐ โ ๐๐๐๐
=๐๐๐๐ โ ๐๐๐๐
I know that if ๐ด๐ด and ๐ต๐ต are both positive and ๐ด๐ด >๐ต๐ต, then ๐ด๐ด
๐ต๐ต> 1. That means if
I divide ๐๐๐๐๐๐๐๐
1512 by ๐๐๐๐
๐๐๐๐
๐๐๐๐๐๐๐๐ and the
quotient is greater than
1, then ๐๐๐๐๐๐๐๐
1512 is larger than ๐๐๐๐
๐๐๐๐
๐๐๐๐๐๐๐๐.
Since ๐ฅ๐ฅ โ 0 and ๐ฅ๐ฅ โ 4, I can divide both sides of the equation by ๐๐
๐๐โ๐๐
without changing their value.
I know the product of two fractions whose values are greater than 1 is greater than 1.
I can define the unknown number as ๐ฆ๐ฆ.
ยฉ 2015 Great Minds eureka-math.org
50
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 25: Adding and Subtracting Rational Expressions
Lesson 25: Adding and Subtracting Rational Expressions
1. Write each sum or difference as a single rational expression.
a. โ๐๐๐๐
+ โ๐๐๐๐
โ๐๐๐๐
+โ๐๐๐๐
=๐๐ โ โ๐๐๐๐ โ ๐๐
+๐๐ โ โ๐๐๐๐ โ ๐๐
=๐๐โ๐๐ + ๐๐โ๐๐
๐๐๐๐
b. ๐๐๐๐+๐๐
+ ๐๐๐๐โ๐๐
๐๐๐๐ + ๐๐
+๐๐
๐๐ โ ๐๐=
๐๐ โ (๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐)
+๐๐(๐๐ + ๐๐)
(๐๐ โ ๐๐)(๐๐ + ๐๐)=๐๐ โ (๐๐ โ ๐๐) + ๐๐(๐๐ + ๐๐)
(๐๐ + ๐๐)(๐๐ โ ๐๐)=๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐
(๐๐ + ๐๐)(๐๐ โ ๐๐)
c. ๐๐๐๐โ๐๐
โ ๐๐๐๐๐๐+๐๐๐๐
โ ๐๐๐๐๐๐โ๐๐๐๐
๐๐๐๐โ ๐๐
โ๐๐
๐๐๐๐ + ๐๐๐๐โ
๐๐๐๐๐๐ โ๐๐๐๐
=๐๐
๐๐โ ๐๐โ
๐๐๐๐(๐๐ + ๐๐)โ
๐๐(๐๐ โ๐๐)(๐๐ +๐๐)
=๐๐(๐๐ + ๐๐)
๐๐(๐๐ + ๐๐)(๐๐โ๐๐)โ๐๐(๐๐โ๐๐)
๐๐(๐๐ + ๐๐)(๐๐โ๐๐) +๐๐๐๐
๐๐(๐๐โ๐๐)(๐๐ +๐๐)
=๐๐(๐๐ + ๐๐) โ (๐๐โ๐๐) + ๐๐๐๐
๐๐(๐๐โ๐๐)(๐๐ + ๐๐) =๐๐๐๐ + ๐๐๐๐
๐๐(๐๐โ๐๐)(๐๐+ ๐๐)
=๐๐(๐๐ + ๐๐)
๐๐(๐๐โ๐๐)(๐๐ + ๐๐)
=๐๐
๐๐(๐๐โ๐๐)
d. ๏ฟฝ๐๐๐๐โ ๐๐
๐๐๏ฟฝ ๏ฟฝ๐๐+๐๐
๐๐+๐๐+ ๐๐โ๐๐
๐๐โ๐๐๏ฟฝ
๏ฟฝ๐๐๐๐โ๐๐๐๐๏ฟฝ ๏ฟฝ๐๐ + ๐ ๐ ๐๐ + ๐๐
+๐๐ โ ๐ ๐ ๐๐ โ ๐๐
๏ฟฝ = ๏ฟฝ๐๐ โ ๐๐๐๐ โ ๐๐
โ๐๐ โ ๐๐๐๐ โ ๐๐
๏ฟฝ๏ฟฝ(๐๐ + ๐ ๐ )(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐) +
(๐๐ โ ๐ ๐ )(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ + ๐๐)๏ฟฝ
=๐๐๐๐ โ ๐๐๐๐
๐๐๐๐ ๏ฟฝ(๐๐ + ๐ ๐ )(๐๐ โ ๐๐) + (๐๐ โ ๐ ๐ )(๐๐ + ๐๐)
๐๐๐๐ โ ๐๐๐๐ ๏ฟฝ
=๐๐๐๐ โ ๐๐๐๐ + ๐๐๐ ๐ โ ๐๐๐ ๐ + ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐ ๐ โ ๐ ๐ ๐๐
๐๐๐๐=๐๐๐๐๐๐ โ ๐๐๐ ๐ ๐๐
๐๐๐๐
This looks almost like the pattern for a square of a difference, but the coefficient of the ๐ฅ๐ฅ2 term is negative, so I cannot factor the expression.
I know (๐๐ โ๐๐) =โ1 โ (๐๐โ ๐๐), so I can multiply the third fraction by โ2
โ2 to
write an equivalent fraction with the least common denominator of 2(๐๐ โ ๐๐)(๐๐ + ๐๐).
ยฉ 2015 Great Minds eureka-math.org
51
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 26: Solving Rational Equations
Lesson 26: Solving Rational Equations
1. Solve the following equations, and check for extraneous solutions.
a. 6๐ฅ๐ฅโ18๐ฅ๐ฅโ3
= 6
๐๐๐๐โ๐๐๐๐๐๐โ๐๐
= ๐๐(๐๐โ๐๐)๐๐โ๐๐
= ๐๐ if ๐๐ โ ๐๐.
Therefore, the equation is true for all real numbers except ๐๐.
b. ๐ฆ๐ฆ+1๐ฆ๐ฆ+3
+ 1๐ฆ๐ฆ
= y+9๐ฆ๐ฆ2+3๐ฆ๐ฆ
First, we multiply both sides of the equation by the least common denominator ๐๐(๐๐ + ๐๐):
๐๐(๐๐ + ๐๐) ๏ฟฝ๐๐+๐๐๐๐+๐๐
๏ฟฝ+ ๐๐(๐๐ + ๐๐) ๏ฟฝ๐๐๐๐๏ฟฝ = ๐๐(๐๐ + ๐๐) ๏ฟฝ ๐๐+๐๐
๐๐(๐๐+๐๐)๏ฟฝ when ๐๐ โ ๐๐ and ๐๐ โ โ๐๐
๐๐(๐๐ + ๐๐) + (๐๐ + ๐๐) = ๐๐ + ๐๐ ๐๐๐๐ + ๐๐ + ๐๐ + ๐๐ = ๐๐ + ๐๐
๐๐๐๐ + ๐๐ โ ๐๐ = ๐๐
The solutions to this equation are โ๐๐ and ๐๐.
Because โ๐๐ is an excluded value, the only solution to the original equation is ๐๐ = ๐๐.
I can multiply each side of the equation by the least common denominator ๐ฆ๐ฆ(๐ฆ๐ฆ + 3). I need to remember to exclude values 0 and โ3 from the possible solutions.
I know that the denominator of the fraction equals 0 when ๐ฅ๐ฅ = 3, which makes the expression undefined for this value.
ยฉ 2015 Great Minds eureka-math.org
52
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 26: Solving Rational Equations
2. Create and solve a rational equation that has 1 as an extraneous solution.
If ๐๐ is an extraneous solution, there must be a factor of (๐๐ โ ๐๐) in the denominator of one or more ofthe rational expressions in the equation. Also, ๐๐ must be a potential solution to the rational equation.
For example, ๐๐ is a solution to ๐๐ + ๐๐ = ๐๐, so the rational equation ๐๐+๐๐๐๐โ๐๐
= ๐๐๐๐โ๐๐
has an extraneous solution of ๐๐.
Check: Suppose that ๐๐+๐๐๐๐โ๐๐
= ๐๐๐๐โ๐๐
. Then (๐๐+๐๐)(๐๐โ๐๐)(๐๐โ๐๐) = ๐๐(๐๐โ๐๐)
(๐๐โ๐๐) for ๐๐ โ ๐๐.
Equating numerators gives:
๐๐๐๐ โ ๐๐ = ๐๐๐๐ โ ๐๐ ๐๐๐๐ โ ๐๐๐๐ + ๐๐ = ๐๐
(๐๐ โ ๐๐)๐๐ = ๐๐ ๐๐ โ ๐๐ = ๐๐
๐๐ = ๐๐
However, ๐๐ = ๐๐ causes division by zero, so ๐๐ is an extraneous solution.
3. Does there exist a pair of consecutive integers whose reciprocals sum to 78? Explain how you know.
If ๐๐ is an integer, ๐๐ + ๐๐ represents the next integer.๐๐๐๐
+ ๐๐๐๐+๐๐
= ๐๐๐๐
๐๐(๐๐)(๐๐+๐๐)๐๐๐๐(๐๐+๐๐)
+ ๐๐(๐๐)(๐๐)๐๐๐๐(๐๐+๐๐)
= ๐๐(๐๐)(๐๐+๐๐)๐๐๐๐(๐๐+๐๐)
for ๐๐ โ ๐๐ and ๐๐ โ โ๐๐
Equating numerators gives ๐๐(๐๐ + ๐๐) + ๐๐๐๐ = ๐๐๐๐(๐๐ + ๐๐)
๐๐๐๐๐๐ + ๐๐ = ๐๐๐๐๐๐ + ๐๐๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐ = ๐๐.
Using the quadratic formula gives
๐๐ = ๐๐ยฑ๏ฟฝ(โ๐๐)๐๐โ๐๐(๐๐)(โ๐๐)๐๐(๐๐)
, which means that the solutions are ๐๐+โ๐๐๐๐๐๐๐๐๐๐
and ๐๐โโ๐๐๐๐๐๐๐๐๐๐
. Thus, there are no
integer solutions.
I can model the relationship in the problem with an equation.
If I write the expressions with a common denominator, I can equate the numerators for all defined values of ๐ฅ๐ฅ and solve the resulting equation.
I know that โ305 is not an integer.
ยฉ 2015 Great Minds eureka-math.org
53
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 27: Word Problems Leading to Rational Equations
Lesson 27: Word Problems Leading to Rational Equations
1. If two air pumps can fill a bounce house in 10 minutes, and one air pump can fill the bounce house in 30minutes on its own, how long would the other air pump take to fill the bounce house on its own?
Let ๐๐ represent the time in minutes it takes the second air pump to fill the bounce house on its own.
๐๐๐๐๐๐
+๐๐๐๐
=๐๐๐๐๐๐
๐๐๐๐๐๐๏ฟฝ๐๐๐๐๐๐
+๐๐๐๐
=๐๐๐๐๐๐๏ฟฝ
So, ๐๐๐๐๐๐๐๐๐๐
+ ๐๐๐๐๐๐๐๐
= ๐๐๐๐๐๐๐๐๐๐
, and then ๐๐ + ๐๐๐๐ = ๐๐๐๐. It follows that ๐๐๐๐ = ๐๐๐๐, so ๐๐ = ๐๐๐๐.
It would take the second air pump ๐๐๐๐ minutes to fill the bounce house on its own.
2. The difference in the average speed of two cars is 12 miles per hour. The slower car takes 2 hours longerto travel 385 miles than the faster car takes to travel 335 miles. Find the speed of the faster car.
Let t represent the time it takes, in hours, for the faster car to drive 335 miles.
๐๐๐๐๐๐๐๐
โ๐๐๐๐๐๐๐๐ + ๐๐
= ๐๐๐๐
๏ฟฝ๐๐๐๐๐๐(๐๐)(๐๐+ ๐๐)
๐๐ ๏ฟฝ โ ๏ฟฝ๐๐๐๐๐๐(๐๐)(๐๐ + ๐๐)
๐๐ + ๐๐ ๏ฟฝ = ๐๐๐๐๐๐(๐๐ + ๐๐)
Thus, ๐๐๐๐๐๐(๐๐ + ๐๐) โ ๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ and ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐. Combining like terms gives ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐, so ๐๐(๐๐ โ ๐๐)(๐๐๐๐ + ๐๐๐๐) = ๐๐.
The only possible solution is ๐๐.
Because ๐๐๐๐๐๐๐๐
= ๐๐๐๐, the speed of the faster car is ๐๐๐๐ miles per hour.
I know multiplying by 30๐ฅ๐ฅ will not change the solution to the equation because ๐ฅ๐ฅ does not equal zero (it would not make sense for an air pump to take 0 minutes to fill the bounce house).
I know the fraction of a bounce house filled by the first air pump in one minute added to the fraction of a bounce house filled by the second air pump in one minute is equal to the fraction of the bounce house filled by both air pumps in one minute.
I know distance is the average speed multiplied by time, so average speed is distance divided by time. I also know that the difference in the average speeds of the cars is 12 miles per hour.
Solving (6๐ก๐ก + 67) = 0 produces a negative solution for ๐ก๐ก, which does not make sense in this context.
Average speed is distance divided by time.
ยฉ 2015 Great Minds eureka-math.org
54
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 27: Word Problems Leading to Rational Equations
I can multiply this equation by the least common denominator ๐๐(๐ก๐ก + 1) because I know neither factor has a value of zero.
3. Consider an ecosystem of squirrels on a college campus that starts with 30 squirrels and can sustain up to150 squirrels. An equation that roughly models this scenario is
๐๐ =150
1 + 4๐ก๐ก + 1
,
where ๐๐ represents the squirrel population in year ๐ก๐ก of the study.
a. Solve this equation for ๐ก๐ก. Describe what the resulting equation represents in the context of thisproblem.
Since ๐ท๐ท = ๐๐๐๐๐๐
๐๐+ ๐๐๐๐+๐๐
, it follows that ๐ท๐ท๏ฟฝ๐๐ + ๐๐๐๐+๐๐
๏ฟฝ = ๐๐๐๐๐๐, and then ๐๐ + ๐๐๐๐+๐๐
= ๐๐๐๐๐๐๐ท๐ท
.
Then
๐ท๐ท(๐๐ + ๐๐) ๏ฟฝ๐๐ +๐๐
๐๐ + ๐๐๏ฟฝ = ๐ท๐ท(๐๐ + ๐๐) ๏ฟฝ
๐๐๐๐๐๐๐ท๐ท
๏ฟฝ
๐ท๐ท(๐๐ + ๐๐) + ๐๐๐ท๐ท = ๐๐๐๐๐๐(๐๐ + ๐๐) ๐ท๐ท๐๐ + ๐ท๐ท + ๐๐๐ท๐ท = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ โ ๐ท๐ท๐๐ = ๐๐๐ท๐ท โ ๐๐๐๐๐๐ ๐๐(๐๐๐๐๐๐ โ ๐ท๐ท) = ๐๐๐ท๐ท โ ๐๐๐๐๐๐
๐๐ =๐๐๐ท๐ท โ ๐๐๐๐๐๐๐๐๐๐๐๐ โ ๐ท๐ท
The resulting equation represents the relationship between the number of squirrels, ๐ท๐ท, and the year, ๐๐. If we know how many squirrels we have, between ๐๐๐๐ and ๐๐๐๐๐๐, we will know how long it took for the squirrel population to grow from ๐๐๐๐ to that value, ๐ท๐ท. If the population is ๐๐๐๐, then this equation says we are in year ๐๐ of the study, which fits with the given scenario.
b. At what time does the population reach 100 squirrels?
When ๐ท๐ท = ๐๐๐๐๐๐, then ๐๐ = ๐๐(๐๐๐๐๐๐)โ๐๐๐๐๐๐๐๐๐๐๐๐โ๐๐๐๐๐๐
= ๐๐๐๐๐๐๐๐๐๐
= ๐๐; therefore, the squirrel population is ๐๐๐๐๐๐ in year ๐๐ of the study.
I need to assume that ๐๐ โ 150 so that I can divide both sides of the equation by 150 โ ๐๐.
ยฉ 2015 Great Minds eureka-math.org
55
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 28: A Focus on Square Roots
Lesson 28: A Focus on Square Roots
Rationalize the Denominator
1. Rationalize the denominator in the following expressions:
a. ๐๐๐๐๐๐โโ๐๐
If ๐๐ โ ๐๐๐๐, then ๐๐๐๐
๐๐ โ โ๐๐=
๐๐๐๐๐๐ โ โ๐๐
โ๐๐ + โ๐๐๐๐ + โ๐๐
=๐๐๐๐(๐๐) + ๐๐๐๐๏ฟฝโ๐๐๏ฟฝ
๐๐(๐๐) + ๐๐โ๐๐ โ ๐๐โ๐๐ โ (โ๐๐)๐๐=๐๐๐๐๐๐ + ๐๐๐๐โ๐๐๐๐๐๐ โ ๐๐
.
b. 3โ๐ฅ๐ฅ+4
If ๐๐ > โ๐๐, then ๐๐
โ๐๐ + ๐๐=
๐๐โ๐๐ + ๐๐
โโ๐๐ + ๐๐โ๐๐ + ๐๐
=๐๐โ๐๐ + ๐๐๐๐ + ๐๐
.
Solve Equations That Contain Radicals
2. Solve each equation, and check the solutions.
a. 3โ๐ฅ๐ฅ + 3 + 4 = โ2
๐๐โ๐๐ + ๐๐ = โ๐๐โ๐๐ + ๐๐ = โ๐๐
๏ฟฝโ๐๐ + ๐๐๏ฟฝ๐๐
= (โ๐๐)๐๐
๐๐ + ๐๐ = ๐๐ ๐๐ = ๐๐
Check: If ๐๐ = ๐๐ is a solution, then โ๐๐ + ๐๐ = โ๐๐, but ๐๐ โ โ๐๐, so ๐๐ is an extraneous solution.
There are no real solutions to the original equation.
b. โ๐ฅ๐ฅ + 5 โ 2โ๐ฅ๐ฅ โ 1 = 0
โ๐๐ + ๐๐ = ๐๐โ๐๐ โ ๐๐
๏ฟฝโ๐๐ + ๐๐๏ฟฝ๐๐
= ๏ฟฝ๐๐โ๐๐ โ ๐๐๏ฟฝ๐๐
๐๐ + ๐๐ = ๐๐(๐๐ โ ๐๐) ๐๐ + ๐๐ = ๐๐๐๐ โ ๐๐ ๐๐๐๐ = ๐๐ ๐๐ = ๐๐
Check: โ๐๐ + ๐๐ โ ๐๐โ๐๐ โ ๐๐ = โ๐๐ โ ๐๐โ๐๐ = ๐๐โ๐๐ โ ๐๐โ๐๐ = ๐๐, so ๐๐ is a valid solution.
I know that โ๐๐ โโ๐๐ = ๐๐ for non-negative values of ๐๐. In this case,
๐๐ = ๐ฅ๐ฅ + 4.
I first need to separate the square root expressions on opposite sides of the equal sign and then square both sides of the equation.
I know that when the denominator of a rational expression is in the form ๐๐ +โ๐๐, I can rationalize the denominator bymultiplying both the numerator and denominator by ๐๐ โ โ๐๐.
I can see that there are no real solutions because the principal square root of a number cannot be negative.
ยฉ 2015 Great Minds eureka-math.org
56
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 28: A Focus on Square Roots
c. โ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ = 6
๏ฟฝ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๏ฟฝ๐๐
= ๐๐๐๐
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ = ๐๐
(๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐ Either ๐๐ = ๐๐ or ๐๐ = โ๐๐
Check: ๏ฟฝ๐๐๐๐ โ ๐๐(๐๐) = โ๐๐๐๐ โ ๐๐๐๐ = โ๐๐๐๐ = ๐๐; ๏ฟฝ(โ๐๐)๐๐ โ ๐๐(โ๐๐) = โ๐๐๐๐ + ๐๐๐๐ = โ๐๐๐๐ = ๐๐
Both ๐๐ and โ๐๐ are solutions to the original equation.
d. 3โ๐ฅ๐ฅโ1
= 2
๐๐โ๐๐โ๐๐
โ (โ๐๐ โ ๐๐) = ๐๐(โ๐๐ โ ๐๐) for ๐๐ โ ๐๐
๐๐ = ๐๐โ๐๐ โ ๐๐(๐๐)๐๐ = ๐๐โ๐๐๐๐๐๐
= โ๐๐ ๐๐๐๐๐๐
= ๐๐
Check: ๐๐
๏ฟฝ๐๐๐๐๐๐ โ๐๐
= ๐๐๐๐๐๐โ๐๐
= ๐๐๏ฟฝ๐๐๐๐๏ฟฝ
= ๐๐ โ ๐๐๐๐ = ๐๐
The only solution is ๐๐๐๐๐๐
.
e. โ2๐ฅ๐ฅ + 13 + 5 = 2
โ๐๐๐๐ + ๐๐๐๐ = โ๐๐ ๏ฟฝโ๐๐๐๐ + ๐๐๐๐ ๏ฟฝ
๐๐= (โ๐๐)๐๐
๐๐๐๐ + ๐๐ = โ๐๐๐๐ ๐๐๐๐ = โ๐๐๐๐ ๐๐ = โ๐๐๐๐
Check: ๏ฟฝ๐๐(โ๐๐๐๐) + ๐๐๐๐ + ๐๐ = โโ๐๐๐๐๐๐ + ๐๐ = โ๐๐ + ๐๐ = ๐๐
The only solution is โ๐๐๐๐.
I could also solve this equation by rationalizing the denominator of 3
โ๐ฅ๐ฅโ1, but multiplying by the least
common denominator takes fewer steps.
ยฉ 2015 Great Minds eureka-math.org
57
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 29: Solving Radical Equations
Lesson 29: Solving Radical Equations
Solve each equation.
1. โ3๐ฅ๐ฅ + 1 = 1 โ 2๐ฅ๐ฅ
๏ฟฝโ๐๐๐๐ + ๐๐๏ฟฝ๐๐
= (๐๐ โ ๐๐๐๐)๐๐
๐๐๐๐ + ๐๐ = ๐๐๐๐ โ ๐๐(๐๐)(๐๐๐๐) + (๐๐๐๐)๐๐ ๐๐๐๐ + ๐๐ = ๐๐ โ ๐๐๐๐ + ๐๐๐๐๐๐
๐๐๐๐๐๐ โ ๐๐๐๐ = ๐๐ ๐๐(๐๐๐๐ โ ๐๐) = ๐๐
๐๐ = ๐๐ or ๐๐ =๐๐๐๐
If ๐๐ = ๐๐, then โ๐๐๐๐ + ๐๐ = ๏ฟฝ๐๐(๐๐) + ๐๐ = ๐๐, and ๐๐ โ ๐๐๐๐ = ๐๐ โ ๐๐ = ๐๐, so ๐๐ is a valid solution.
If ๐๐ = ๐๐๐๐, then โ๐๐๐๐ + ๐๐ = ๏ฟฝ๐๐๏ฟฝ๐๐
๐๐๏ฟฝ + ๐๐ = ๏ฟฝ๐๐๐๐
๐๐= ๐๐
๐๐, and ๐๐ โ ๐๐๐๐ = ๐๐ โ ๐๐๐๐
๐๐= โ๐๐
๐๐, so ๐๐
๐๐ is not a solution.
The only solution is ๐๐.
2. โ๐ฅ๐ฅ โ 5 + โ2๐ฅ๐ฅ โ 3 = 4
โ๐๐ โ ๐๐ = ๐๐ โ โ๐๐๐๐ โ ๐๐
๏ฟฝโ๐๐ โ ๐๐๏ฟฝ๐๐
= ๏ฟฝ๐๐ โ โ๐๐๐๐ โ ๐๐๏ฟฝ๐๐
๐๐ โ ๐๐ = ๐๐๐๐ โ ๐๐(๐๐)โ๐๐๐๐ โ ๐๐ + ๏ฟฝโ๐๐๐๐ โ ๐๐๏ฟฝ๐๐
๐๐ โ ๐๐ = ๐๐๐๐ โ ๐๐โ๐๐๐๐ โ ๐๐ + ๐๐๐๐ โ ๐๐ ๐๐ โ ๐๐ = ๐๐๐๐ + ๐๐๐๐ โ ๐๐โ๐๐๐๐ โ ๐๐ ๐๐ + ๐๐๐๐ = ๐๐โ๐๐๐๐ โ ๐๐
(๐๐ + ๐๐๐๐)๐๐ = ๏ฟฝ๐๐โ๐๐๐๐ โ ๐๐๏ฟฝ๐๐
๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐(๐๐๐๐ โ ๐๐) ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐ (๐๐ โ ๐๐)(๐๐ โ ๐๐๐๐) = ๐๐
๐๐ = ๐๐ or ๐๐ = ๐๐๐๐
If ๐๐ = ๐๐, then โ๐๐ โ ๐๐ + โ๐๐๐๐ โ ๐๐ = โ๐๐ โ ๐๐ + ๏ฟฝ๐๐(๐๐) โ ๐๐ = โ๐๐ + โ๐๐ = ๐๐, so ๐๐ is a solution. If ๐๐ = ๐๐๐๐, then โ๐๐ โ ๐๐ + โ๐๐๐๐ โ ๐๐ = โ๐๐๐๐ โ ๐๐ + ๏ฟฝ๐๐(๐๐๐๐) โ ๐๐ = โ๐๐๐๐ + โ๐๐๐๐๐๐ = ๐๐๐๐ โ ๐๐, so ๐๐๐๐ is not a solution to the original equation.
The only solution to the original equation is ๐๐.
I can use the pattern (๐๐ โ ๐๐)2 = ๐๐2 โ 2๐๐๐๐ + ๐๐2
to expand the expression ๏ฟฝ4 โ โ2๐ฅ๐ฅ โ 3๏ฟฝ2
. I willbe left with a term containing a square root, which I will need to isolate.
I know that squaring an equation does not always generate an equivalent equation, so I will need to check for extraneous solutions possibly introduced by squaring.
Now that I have isolated the radical term, I can square both sides of the equation, which will create a quadratic equation I can solve.
ยฉ 2015 Great Minds eureka-math.org
58
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 29: Solving Radical Equations
3. Consider the trapezoid ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด shown to the right, where ๐ด๐ด๐ด๐ด = ๐ด๐ด๐ด๐ด. a. If the length of ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is 12, the length of ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is 18,
and the height of the trapezoid is ๐ฅ๐ฅ, write an expressionfor the perimeter of trapezoid ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด in terms of ๐ฅ๐ฅ.
(๐จ๐จ๐จ๐จ)๐๐ = (๐จ๐จ๐ซ๐ซ)๐๐ + (๐จ๐จ๐ซ๐ซ)๐๐ (๐จ๐จ๐จ๐จ)๐๐ = ๐๐๐๐ + ๐๐๐๐
๐จ๐จ๐จ๐จ = ๏ฟฝ๐๐ + ๐๐๐๐
Since ๐จ๐จ๐จ๐จ + ๐จ๐จ๐ฉ๐ฉ+ ๐ฉ๐ฉ๐จ๐จ+ ๐จ๐จ๐จ๐จ = ๐๐๐๐ + ๏ฟฝ๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๏ฟฝ๐๐ + ๐๐๐๐, the perimeter of ๐จ๐จ๐จ๐จ๐ฉ๐ฉ๐จ๐จ is ๐๐๐๐ +๐๐๏ฟฝ๐๐ + ๐๐๐๐.
b. Find the value of ๐ฅ๐ฅ for which the perimeter of trapezoid ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด is equal to 60 cm.
๐๐๐๐ + ๐๐๏ฟฝ๐๐ + ๐๐๐๐ = ๐๐๐๐
๐๐๏ฟฝ๐๐ + ๐๐๐๐ = ๐๐๐๐ ๏ฟฝ๐๐ + ๐๐๐๐ = ๐๐๐๐ ๐๐ + ๐๐๐๐ = ๐๐๐๐๐๐ ๐๐ + ๐๐๐๐ = ๐๐๐๐๐๐
๐๐๐๐ = ๐๐๐๐๐๐ ๐๐ = โ๐๐๐๐๐๐ = ๐๐โ๐๐
The height of trapezoid ๐จ๐จ๐จ๐จ๐ฉ๐ฉ๐จ๐จ is ๐๐โ๐๐ centimeters when the perimeter is ๐๐๐๐ centimeters.
I can use the Pythagorean theorem to find the length ๐ด๐ด๐ด๐ด, which has the same value as ๐ด๐ด๐ด๐ด.
Since ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด is an isosceles trapezoid, I can decompose it into a rectangle of width 12 and height ๐ฅ๐ฅ, and two congruent right triangles.
ยฉ 2015 Great Minds eureka-math.org
59
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 30: Linear Systems in Three Variables
Lesson 30: Linear Systems in Three Variables
1. Solve the following systems of equations:a. ๐๐ = 3(๐ ๐ + ๐ก๐ก)
๐ก๐ก = ๐ ๐ โ ๐๐๐๐ = 2๐ ๐ โ 1
๐๐๐๐ โ ๐๐ = ๐๐(๐๐ + ๐๐) ๐๐ = ๐๐ โ (๐๐๐๐ โ ๐๐)
โ๐๐ โ ๐๐๐๐ = ๐๐ ๐๐ + ๐๐ = ๐๐
Adding the equations gives โ๐๐๐๐ = ๐๐, so ๐๐ = โ๐๐.
Then, ๐๐ + ๐๐ = ๐๐ so ๐๐ + (โ๐๐) = ๐๐ and then ๐๐ = ๐๐. Since ๐๐ = ๐๐๐๐ โ ๐๐, it follows that ๐๐ = ๐๐(๐๐) โ ๐๐ and then ๐๐ = ๐๐.
Solution: (๐๐,๐๐,โ๐๐)
b. ๐๐ + ๐๐ + 3๐๐ = 05๐๐ โ 2๐๐ + 3๐๐ = 7๐๐ โ ๐๐ + ๐๐ = 2
Then, ๐๐ = ๐๐. Since ๐๐๐๐ + ๐๐๐๐ = ๐๐ and ๐๐ = ๐๐, it follows that ๐๐๐๐ + ๐๐ = ๐๐ and ๐๐ = ๐๐. Then since ๐๐ + ๐๐ + ๐๐๐๐ = ๐๐, it follows that ๐๐ + ๐๐ + ๐๐(๐๐) = ๐๐ and then ๐๐ = โ๐๐.
Solution: (๐๐,โ๐๐,๐๐)
โ๐๐(๐๐๐๐+ ๐๐๐๐) = โ๐๐(๐๐) ๐๐(๐๐๐๐ + ๐๐๐๐) = ๐๐(๐๐)
โ๐๐๐๐๐๐โ ๐๐๐๐๐๐ = โ๐๐๐๐ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐
If I substitute the expression (2๐ ๐ โ 1) for ๐๐ into the first and second equations, I can create two equations with two variables. Since the coefficients of ๐ ๐ are opposites in the resulting equations, I can then add the equations to eliminate ๐ ๐ .
Once I find the value of ๐ก๐ก, I can back substitute its value into ๐ ๐ + ๐ก๐ก = 1 to find the value of ๐ ๐ and then back substitute the value of ๐ ๐ into ๐๐ = 2๐ ๐ โ 1 to find the value of ๐๐.
If I add the first and third equations, I can eliminate ๐๐. I can also eliminate ๐๐ by adding twice the first equation to the second equation.
I can add these equations to solve for ๐๐.
Multiply both sides of ๐๐๐๐ + ๐๐๐๐ = ๐๐ by โ๐๐, and multiply both sides of ๐๐๐๐ + ๐๐๐๐ = ๐๐ by ๐๐.
๐๐ + ๐๐ + ๐๐๐๐ = ๐๐ + ๐๐ โ ๐๐ + ๐๐ = ๐๐ ๐๐๐๐ + ๐๐๐๐ = ๐๐
๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ = ๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐ ๐๐๐๐ + ๐๐๐๐ = ๐๐
โ๐๐๐๐๐๐โ ๐๐๐๐๐๐ = โ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐
โ๐๐๐๐๐๐ = ๐๐
ยฉ 2015 Great Minds eureka-math.org
60
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 30: Linear Systems in Three Variables
๐๐ + ๐๐ โ ๐๐ = ๐๐ + ๐๐ + ๐๐ + ๐๐ = ๐๐ ๐๐๐๐ + ๐๐๐๐ = ๐๐
๐๐ + ๐๐ + ๐๐ = ๐๐ + ๐๐๐๐ โ ๐๐ = ๐๐ ๐๐๐๐ + ๐๐ = ๐๐
๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ โ ๐๐ = โ๐๐
๐๐๐๐ = ๐๐
c. 1๐ฅ๐ฅ
+ 1๐ฆ๐ฆโ 1
๐ง๐ง= 0
3๐ฅ๐ฅโ
1๐ง๐ง
= 0
1๐ฅ๐ฅ
+1๐ฆ๐ฆ
+1๐ง๐ง
= 6
Let ๐๐ = ๐๐๐๐, ๐๐ = ๐๐
๐๐, ๐๐ = ๐๐
๐๐. Then the system becomes
๐๐ + ๐๐ โ ๐๐ = ๐๐ ๐๐๐๐ โ ๐๐ = ๐๐ ๐๐ + ๐๐ + ๐๐ = ๐๐
So, ๐๐ = ๐๐. Since ๐๐๐๐ + ๐๐ = ๐๐ and ๐๐ = ๐๐, we see that ๐๐ = ๐๐.
Then ๐๐๐๐ โ ๐๐ = ๐๐ and ๐๐ = ๐๐ gives ๐๐ = ๐๐.
Since ๐๐ = ๐๐๐๐ and ๐๐ = ๐๐, we know ๐๐ = ๐๐.
Since ๐๐ = ๐๐๐๐
and ๐๐ = ๐๐, we know ๐๐ = ๐๐๐๐.
Since ๐๐ = ๐๐๐๐ and ๐๐ = ๐๐, we know ๐๐ = ๐๐
๐๐.
Solution: ๏ฟฝ๐๐, ๐๐๐๐
, ๐๐๐๐๏ฟฝ
If I define variables to represent 1๐ฅ๐ฅ
, 1๐ฆ๐ฆ
, and 1๐ง๐ง, I
can rewrite the system of equations in a format similar to that of the other systems I have solved.
I need to be sure to write the solution to the original system by finding values of ๐ฅ๐ฅ,๐ฆ๐ฆ, and ๐ง๐ง.
I can add pairs of equations to eliminate ๐๐.
I can combine the new equations to eliminate ๐๐.
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Lesson 30: Linear Systems in Three Variables
๐๐ + ๐๐๐๐ = ๐๐๐๐ โ ๐๐ โ ๐๐ = โ๐๐๐๐
๐๐ = ๐๐๐๐
2. A chemist needs to make 50 ml of a 12% acid solution. If he uses 10% and 20% solutions to create the12% solution, how many ml of each does he need?
Let ๐๐ represent the amount of ๐๐๐๐% solution used and ๐๐ represent the amount of ๐๐๐๐% solution used.
๐๐ + ๐๐ = ๐๐๐๐ ๐๐.๐๐๐๐ + ๐๐.๐๐๐๐ = ๐๐.๐๐๐๐(๐๐๐๐)
Multiply the second equation by ๐๐๐๐: ๐๐๐๐(๐๐.๐๐๐๐ + ๐๐.๐๐๐๐) = ๐๐๐๐(๐๐.๐๐๐๐)(๐๐๐๐) becomes ๐๐ + ๐๐๐๐ = ๐๐๐๐.
The sum of the amount of the two solutions used is 50 ml.
I can add โ1 times the first equation to ten times the second equation to find the value of ๐ฆ๐ฆ .
The amount of acid in a solution is the product of the volume of the solution and its concentration.
Since ๐๐ = ๐๐๐๐ and ๐๐ + ๐๐ = ๐๐๐๐, we have ๐๐ = ๐๐๐๐. The chemist used ๐ฆ๐ฆ๐ฆ๐ฆ of the ๐๐๐๐% solution and ๐ฆ๐ฆ๐ฆ๐ฆ of the ๐๐๐๐% solution.
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๐๐๐๐ ๐๐๐๐
ALG II-M1-HWH-1.3.0-08.2015
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Lesson 31: Systems of Equations
Lesson 31: Systems of Equations
1. Where do the lines given by ๐ฆ๐ฆ = ๐ฅ๐ฅ โ ๐๐ and ๐ฆ๐ฆ = 3๐ฅ๐ฅ + 1 intersect?
๐๐ โ ๐๐ = ๐๐๐๐ + ๐๐๐๐๐๐ = โ๐๐ โ ๐๐
๐๐ =โ๐๐ โ ๐๐
๐๐๐๐ = ๐๐ โ ๐๐
๐๐ =โ๐๐ โ ๐๐
๐๐โ ๐๐ =
โ๐๐โ ๐๐ โ ๐๐๐๐๐๐
=โ๐๐๐๐โ ๐๐
๐๐
The point of intersection for the lines is ๏ฟฝโ๐๐โ๐๐๐๐
, โ๐๐๐๐โ๐๐๐๐
๏ฟฝ.
2. Find all solutions to the following systems of equations. Illustrate each system with a graph.a. (๐ฅ๐ฅ โ 1)2 + (๐ฆ๐ฆ + 2)2 = 9
2๐ฅ๐ฅ โ ๐ฆ๐ฆ = 7
First, ๐๐๐๐ โ ๐๐ = ๐๐, so ๐๐ = ๐๐๐๐ โ ๐๐.
Then (๐๐ โ ๐๐)๐๐ + ๏ฟฝ(๐๐๐๐ โ ๐๐) + ๐๐๏ฟฝ๐๐ = ๐๐, so (๐๐ โ ๐๐)๐๐ + (๐๐๐๐ โ ๐๐)๐๐ = ๐๐.
๐๐๐๐ โ ๐๐๐๐ + ๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ = ๐๐ (๐๐๐๐ โ ๐๐๐๐)(๐๐ โ ๐๐) = ๐๐
Then ๐๐ = ๐๐๐๐๐๐
or ๐๐ = ๐๐.
If ๐๐ = ๐๐๐๐๐๐
, ๐๐ = ๐๐๏ฟฝ๐๐๐๐๐๐๏ฟฝ โ ๐๐ = โ๐๐
๐๐.
If ๐๐ = ๐๐, ๐๐ = ๐๐(๐๐) โ ๐๐ = โ๐๐.
The points of intersection are ๏ฟฝ๐๐๐๐๐๐
,โ๐๐๐๐๏ฟฝ and (๐๐,โ๐๐).
I can solve the linear equation for ๐ฆ๐ฆ and substitute the result into the quadratic equation.
I should back substitute these values of ๐ฅ๐ฅ into the linear equation 2๐ฅ๐ฅ โ ๐ฆ๐ฆ = 7 to avoid extraneous solutions that might arise if I used the more complicated equation.
I know that, at the point of intersection, the ๐ฆ๐ฆ-coordinates will be the same, so I can set ๐ฅ๐ฅ โ ๐๐ = 3๐ฅ๐ฅ + 1.
My graph shows the two points of intersection.
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Lesson 31: Systems of Equations
b. ๐ฆ๐ฆ = 4๐ฅ๐ฅ โ 11๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 5
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐ (๐๐ โ ๐๐)(๐๐ โ ๐๐) = ๐๐
The only solution is ๐๐.
If ๐๐ = ๐๐, then ๐๐ = ๐๐(๐๐) โ ๐๐๐๐ = ๐๐.
The only point of intersection is (๐๐,๐๐).
3. Find all values of ๐๐ so that the following system has nosolutions.(๐ฆ๐ฆ โ ๐๐)2 = ๐ฅ๐ฅ + 3๐ฆ๐ฆ = โ2๐ฅ๐ฅ + ๐๐
๏ฟฝ(โ๐๐๐๐+ ๐๐) โ ๐๐๏ฟฝ๐๐ = ๐๐ + ๐๐(โ๐๐๐๐)๐๐ = ๐๐ + ๐๐ ๐๐๐๐๐๐ โ ๐๐ โ ๐๐ = ๐๐
(๐๐๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐
๐๐ = ๐๐ or ๐๐ = โ๐๐๐๐
If ๐๐ = ๐๐, then ๐๐ = โ๐๐(๐๐) + ๐๐ = ๐๐ โ ๐๐.
If ๐๐ = โ๐๐๐๐, then ๐๐ = โ๐๐๏ฟฝโ ๐๐
๐๐๏ฟฝ + ๐๐ = ๐๐ + ๐๐
๐๐, so the intersection points are (๐๐,๐๐ โ ๐๐) and ๏ฟฝโ๐๐
๐๐,๐๐ + ๐๐
๐๐๏ฟฝ.
The graphs will intersect in two locations regardless of the value of ๐๐. There are no values for ๐๐ so that the system has no solutions.
The repeated factor indicates that there is one point of intersection, which is shown on the graph.
These are examples of the graphs of systems for two different values of ๐๐.
Substitute โ2๐ฅ๐ฅ + ๐๐ for ๐ฆ๐ฆ in the first equation, and solve for ๐ฅ๐ฅ.
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Lesson 32: Graphing Systems of Equations
If the graph of the circle does not contain the point (0, 3), either the circle does not intersect the ๐ฆ๐ฆ-axis, or it intersects the ๐ฆ๐ฆ-axis in two places.
Lesson 32: Graphing Systems of Equations
1. Use the distance formula to find the length of the diagonal of a rectangle whose vertices are (1,3), (4,3),(4,9), and (1,9).
Using the vertices (๐๐,๐๐) and (๐๐,๐๐), we have ๐ ๐ = ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐ = โ๐๐ + ๐๐๐๐ = โ๐๐๐๐ = ๐๐โ๐๐.
The diagonal is ๐๐โ๐๐ units long.
2. Write an equation for the circle with center (2, 3) in the form (๐ฅ๐ฅ โ โ)2 + (๐ฆ๐ฆ โ ๐๐)2 = ๐๐2 that is tangentto the ๐ฆ๐ฆ-axis, where the center is (โ,๐๐) and the radius is ๐๐. Then write the equation in the standard form๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐ฆ๐ฆ2 + ๐๐๐ฆ๐ฆ + ๐๐ = 0, and construct the graph of the equation.
Since the center of the circle has coordinates (๐๐,๐๐) and the circle istangent at to the ๐๐-axis, the circle must pass through the point (๐๐,๐๐),which means the radius is ๐๐ units long.
Equation for circle: (๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐ = ๐๐๐๐
(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐ = ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ + ๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐ = ๐๐
Standard form: ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐ = ๐๐
Since the diagonals of a rectangle are congruent, I can use any two nonadjacent vertices to find the length of the diagonal.
I know ๐๐ = ๏ฟฝ(๐ฅ๐ฅ2 โ ๐ฅ๐ฅ1)2 + (๐ฆ๐ฆ2 โ ๐ฆ๐ฆ1)2 by the distance formula.
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Lesson 32: Graphing Systems of Equations
I can substitute (๐ฅ๐ฅ โ 1) for ๐ฆ๐ฆ2 in the second equation.
I can use the distance formula to find the distance between the centers (0, 0) and (3,โ3).
I know that if two circles intersect, the distance between the centers is less than or equal to the sum of their radii. Since the distance between the centers is greater than the sum of the radii, the circles cannot overlap.
3. By finding the radius of each circle and the distance between their centers, show that the circles withequations ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 = 1 and ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + ๐ฆ๐ฆ2 + 6๐ฆ๐ฆ + 9 = 0 do not intersect. Illustrate graphically.
The graph of ๐๐๐๐ + ๐๐๐๐ = ๐๐ has center (๐๐,๐๐) and radius ๐๐ unit.
๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐ = ๐๐ ๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๏ฟฝ + ๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐
(๐๐ โ ๐๐)๐๐ + (๐๐ + ๐๐)๐๐ = ๐๐
The graph of ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐ = ๐๐ is a circle with center (๐๐,โ๐๐) and radius 3 units.
Since ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (โ๐๐ โ ๐๐)๐๐ = โ๐๐๐๐, the distance between the centers of the circles is โ๐๐๐๐ units.
Since ๐๐ + ๐๐ = ๐๐, the sum of the radii is ๐๐ units. Since โ๐๐๐๐ > ๐๐, the circles do not overlap.
4. Solve the system ๐ฅ๐ฅ = ๐ฆ๐ฆ2 + 1 and ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 = 5.Illustrate graphically.
Because ๐๐ = ๐๐๐๐ + ๐๐, it follows that ๐๐๐๐ = ๐๐ โ ๐๐.
If ๐๐ = ๐๐, then ๐๐๐๐ = ๐๐ โ ๐๐, so ๐๐๐๐ = ๐๐. Thus either ๐๐ = ๐๐ or ๐๐ = โ๐๐. If ๐๐ = โ๐๐, then ๐๐๐๐ = โ๐๐ โ ๐๐, so ๐๐๐๐ = โ๐๐ which has no real solution.
The points of intersection are (๐๐,๐๐) and (๐๐,โ๐๐).
๐๐๐๐ + ๐๐๐๐ = ๐๐ ๐๐๐๐ + (๐๐ โ ๐๐) = ๐๐
๐๐๐๐ + ๐๐ โ ๐๐ โ ๐๐ = ๐๐ ๐๐๐๐ + ๐๐ โ ๐๐ = ๐๐
(๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐ ๐๐ = ๐๐ or ๐๐ = โ๐๐
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Lesson 33: The Definition of a Parabola
Lesson 33: The Definition of a Parabola
1. Demonstrate your understanding of the definition of a parabola by drawing several pairs of congruentsegments given the parabola, its focus, and directrix. Measure the segments that you drew in eitherinches or centimeters to confirm the accuracy of your sketches.
2. Find the values of ๐ฅ๐ฅ for which the point (๐ฅ๐ฅ, 2) is equidistant from (3,5) and the line ๐ฆ๐ฆ = โ3.
The distance from (๐๐,๐๐) to (๐๐,๐๐) is ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐ + ๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐.
Then, ๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐ ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐ = ๐๐
(๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐ ๐๐ = ๐๐ or ๐๐ = โ๐๐
The points (๐๐,๐๐) and (โ๐๐,๐๐) are both equidistant from the point (๐๐,๐๐) and the line ๐๐ = โ๐๐.
3. Consider the equation ๐ฆ๐ฆ = 14๐ฅ๐ฅ2 + 2๐ฅ๐ฅ.
a. Find the coordinates of the points on the graph of ๐ฆ๐ฆ = 14๐ฅ๐ฅ2 + 2๐ฅ๐ฅ whose ๐ฅ๐ฅ-values are 0 and 2.
If ๐๐ = ๐๐, then ๐๐ = ๐๐๐๐
(๐๐)๐๐ + ๐๐(๐๐) = ๐๐.
If ๐๐ = ๐๐, then ๐๐ = ๐๐๐๐
(๐๐)๐๐ + ๐๐(๐๐) = ๐๐.
The coordinates are (๐๐,๐๐) and (๐๐,๐๐).
I know the distance between a point (๐๐, ๐๐) and a horizontal line ๐ฆ๐ฆ = ๐๐ is |๐๐ โ ๐๐|.
I can set the distances I calculated equal to each other.
Because ๐๐ โ (โ๐๐) = ๐๐, the distance from(๐๐,๐๐) to the line ๐๐ = โ๐๐ is ๐๐ units.
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Lesson 33: The Definition of a Parabola
b. Show that each point in part (a) is equidistant from the point (โ4,โ3) and the line ๐ฆ๐ฆ = โ5.
The distance from the point (๐๐,๐๐) to the line ๐๐ = โ๐๐ is ๐๐.
The distance between (๐๐,๐๐) and (โ๐๐,โ๐๐) is ๏ฟฝ(โ๐๐โ ๐๐)๐๐ + (โ๐๐ โ ๐๐)๐๐ = โ๐๐๐๐ = ๐๐.
Thus, the point (๐๐,๐๐) is equidistant from the point (โ๐๐,โ๐๐) and the line ๐๐ = โ๐๐.
The distance from the point (๐๐,๐๐) to the line ๐๐ = โ๐๐ is ๐๐๐๐.
The distance between (๐๐,๐๐) and (โ๐๐,โ๐๐) is ๏ฟฝ(โ๐๐โ ๐๐)๐๐ + (โ๐๐ โ ๐๐)๐๐ = โ๐๐๐๐๐๐ = ๐๐๐๐.
Thus, the point (๐๐,๐๐) is equidistant from the point (โ๐๐,โ๐๐) and the line ๐๐ = โ๐๐.
c. Show that if the point with coordinates (๐ฅ๐ฅ,๐ฆ๐ฆ) is equidistant from the point (โ4,โ3) and the line๐ฆ๐ฆ = โ5, then ๐ฆ๐ฆ = 1
4๐ฅ๐ฅ2 + 2๐ฅ๐ฅ.
The distance between the point with coordinates (๐๐,๐๐) and the line ๐๐ = โ๐๐ is ๐๐+ ๐๐.
The distance between the points (๐๐,๐๐) and ๏ฟฝโ๐๐,โ๐๐๏ฟฝ is ๏ฟฝ(๐๐ + ๐๐)๐๐ + (๐๐ + ๐๐)๐๐.
So, if ๐๐ + ๐๐ = ๏ฟฝ(๐๐ + ๐๐)๐๐ + (๐๐ + ๐๐)๐๐, then (๐๐ + ๐๐)๐๐ = (๐๐ + ๐๐)๐๐ + (๐๐ + ๐๐)๐๐
๐๐๐๐ + ๐๐๐๐๐๐+ ๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐
๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐
๐๐ =๐๐๐๐ ๏ฟฝ๐๐๐๐ + ๐๐๐๐๏ฟฝ
๐๐ =๐๐๐๐๐๐๐๐ + ๐๐๐๐
4. Derive the analytic equation of a parabola with focus (2,6) and directrix on the ๐ฅ๐ฅ-axis.
The distance between the point with coordinates (๐๐,๐๐) and the line ๐๐ = ๐๐ is ๐๐, and the distance betweenthe points (๐๐,๐๐) and (๐๐,๐๐) is ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐. These distances are equal.
๐๐ = ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐
๐๐๐๐ = (๐๐ โ ๐๐)๐๐ + ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ ๐๐๐๐ โ ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐๏ฟฝ = (๐๐ โ ๐๐)๐๐ + ๐๐๐๐
๐๐๐๐๐๐ = (๐๐ โ ๐๐)๐๐ + ๐๐๐๐
๐๐ =๐๐๐๐๐๐ ๏ฟฝ
(๐๐ โ ๐๐)๐๐ + ๐๐๐๐๏ฟฝ
๐๐ =๐๐๐๐๐๐
(๐๐ โ ๐๐)๐๐ + ๐๐
Since the point (๐ฅ๐ฅ,๐ฆ๐ฆ) is equidistant from (โ4,โ3) and the line, I set these distances equal to find the relationship between ๐ฆ๐ฆ and ๐ฅ๐ฅ.
I know the equation of the parabola will have the form ๐ฆ๐ฆ = ๐๐(๐ฅ๐ฅ โ โ)2 + ๐๐, so I am leaving the expression (๐ฅ๐ฅ โ 2)2 in factored form and expanding the expression (๐ฆ๐ฆ โ 6)2.
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Lesson 34: Are All Parabolas Congruent?
Lesson 34: Are All Parabolas Congruent?
1. Use the definition of a parabola to sketch the parabola defined by the given focus and directrix. Thenwrite an analytic equation for each parabola.a. Focus: (0,4) Directrix: ๐ฆ๐ฆ = 2
The vertex of the parabola is (๐๐,๐๐).
Since the distance from the vertex to the focus is ๐๐ unit, ๐๐
๐๐๐๐ = ๐๐, and then ๐๐ = ๐๐.
The parabola opens up, so ๐๐ = ๐๐๐๐๐๐
(๐๐ โ ๐๐)๐๐ + ๐๐.
Then ๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐.
b. Focus: (4,0) Directrix: ๐ฅ๐ฅ = โ2
The vertex of the parabola is (๐๐,๐๐).
Since the distance from the vertex to the focus is ๐๐ units, ๐๐๐๐๐๐ =
๐๐,and then ๐๐ = ๐๐.
The parabola opens to the right, so ๐๐ = ๐๐๐๐๐๐
(๐๐ โ ๐๐)๐๐ + ๐๐,
which gives ๐๐ = ๐๐๐๐๐๐๐๐๐๐ + ๐๐.
The vertex lies along the ๐ฆ๐ฆ-axis, halfway between the focus and directrix.
I know the distance from the vertex to the focus is 1
2๐๐.
I know the parabola opens upward because the directrix is beneath it. Therefore, I can use the general form for the equation of a parabola that opens up with vertex (โ,๐๐).
I know the parabola opens to the right because the directrix is to the left of it. Because of this, I can use the general form for the equation of a parabola that opens to the right with vertex (โ,๐๐).
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Lesson 34: Are All Parabolas Congruent?
c. Explain how you can tell whether the parabolas in parts (a) and (b) are congruent.
The parabolas are not congruent because they do not have the same value of ๐๐.
2. Let ๐๐ be the parabola with focus (0,8) and directrix ๐ฆ๐ฆ = ๐ฅ๐ฅ.
a. Sketch this parabola. Then write an equation in the form ๐ฆ๐ฆ = 12๐๐๐ฅ๐ฅ2 whose graph is a parabola that is
congruent to ๐๐.
The vertex of the parabola is equidistant from the focus anddirectrix along the line through the focus that isperpendicular to the directrix. The equation of that line is๐๐ โ ๐๐ = โ๐๐(๐๐ โ ๐๐), which simplifies to ๐๐ = โ๐๐ + ๐๐.
The point where the line ๐๐ = โ๐๐ + ๐๐ intersects the directrix has coordinates (๐๐,๐๐), so ๐๐ = โ๐๐ + ๐๐.
Then ๐๐๐๐ = ๐๐, so ๐๐ = ๐๐, and the intersection point on the directrix is (๐๐,๐๐).
The vertex is the point halfway between (๐๐,๐๐) and (๐๐,๐๐), which is ๏ฟฝ๐๐+๐๐๐๐
, ๐๐+๐๐๐๐๏ฟฝ = (๐๐,๐๐).
The distance from the vertex to the focus is ๏ฟฝ(๐๐ โ ๐๐)๐๐ + (๐๐ โ ๐๐)๐๐ = โ๐๐ + ๐๐ = ๐๐โ๐๐.
Then, ๐๐โ๐๐ = ๐๐๐๐๐๐, so ๐๐ = ๐๐โ๐๐. It follows that ๐๐ = ๐๐
๐๐โ๐๐๐๐๐๐.
I know this line will have slope โ1 because that is the slope of a line perpendicular to a line of slope 1, such as ๐ฆ๐ฆ = ๐ฅ๐ฅ.
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Lesson 34: Are All Parabolas Congruent?
The distance between the origin and the point on the directrix that aligns with the focus is 4โ2, so the directrix, vertex, and focus will align at ๐ฅ๐ฅ = 4โ2. The distance from the directrix to the focus is 4โ2.
b. Write an equation whose graph is ๐๐โฒ, the parabola congruent to ๐๐ that results after ๐๐ is rotatedclockwise 45ยฐ about the focus.
The equation for the directrix is ๐๐ = ๐๐ โ ๐๐โ๐๐.The equation for the parabola is ๐๐ = ๐๐
๐๐โ๐๐๐๐๐๐ + ๐๐ โ ๐๐โ๐๐.
c. Write an equation whose graph is ๐๐โฒโฒ, the parabola congruent to ๐๐ that results after the directrix of๐๐ is rotated 45ยฐ clockwise about the origin.
The focus is ๏ฟฝ๐๐โ๐๐,๐๐โ๐๐๏ฟฝ, and the directrix is the ๐๐-axis.
The equation for the parabola is
๐๐ =๐๐
๐๐โ๐๐๏ฟฝ๐๐ โ ๐๐โ๐๐๏ฟฝ
๐๐+ ๐๐โ๐๐.
I know the directrix will be a horizontal line, and the distance between the focus, whose coordinates are (0, 8) and the directrix along the ๐ฆ๐ฆ-axis, is 4โ2.
The distance from the vertex to the focus is 2โ2.
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Lesson 35: Are All Parabolas Similar?
Lesson 35: Are All Parabolas Similar?
1. Let ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ2. The graph of ๐๐ is shown below.a. On the same axes, graph the function ๐๐, where
๐๐(๐ฅ๐ฅ) = ๐๐ ๏ฟฝ12๐ฅ๐ฅ๏ฟฝ. Then, graph the function โ, where โ(๐ฅ๐ฅ) = 2๐๐(๐ฅ๐ฅ).
b. Based on your work, make a conjecture about the resulting function when the original function istransformed with a horizontal scaling and then a vertical scaling by the same factor, ๐๐.
In this example, the resulting function is a dilation with scale factor ๐๐.
2. Let ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2.a. What are the focus and directrix of the parabola that is the graph of the function ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2?
Since ๐๐๐๐๐๐
= ๐๐, we know that the distance between the focus and the directrix is ๐๐ = ๐๐๐๐. The point
(๐๐,๐๐) is both the vertex of the parabola and the midpoint of the segment connecting the focus and the directrix. Since the distance between the focus and vertex is ๐๐
๐๐๐๐, this distance is ๐๐
๐๐, which is the
same as the distance between the vertex and directrix. Therefore, the focus has coordinates ๏ฟฝ๐๐, ๐๐๐๐๏ฟฝ
and the directrix has equation ๐๐ = โ๐๐๐๐.
I notice that applying a horizontal scaling with factor 2 and then applying a vertical scaling with factor 2 to the resulting image produces a graph that is a dilation of the graph of ๐๐ with scale factor 2.
I notice that this equation is in the form ๐ฆ๐ฆ = 12๐๐๐ฅ๐ฅ2, which
produces a parabola with vertex at the origin that opens upward.
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Lesson 35: Are All Parabolas Similar?
b. Describe the sequence of transformations that would take the graph of ๐๐ to each parabola describedbelow. Then write an analytic equation for each parabola described.
i. Focus: ๏ฟฝ0,โ 18๏ฟฝ, directrix: ๐ฆ๐ฆ = 1
8
๐๐ = โ๐๐๐๐๐๐
ii. Focus: (0,โ38), directrix: ๐ฆ๐ฆ = โ 5
8
This parabola is a vertical translation of the graph of ๐๐ down ๐๐๐๐ unit.
๐๐ = ๐๐๐๐๐๐ โ๐๐๐๐
iii. Focus: (1,0), directrix: ๐ฅ๐ฅ = โ1
๐๐ =๐๐๐๐๐๐๐๐
c. Which of the graphs represent parabolas that are similar? Which represent parabolas that arecongruent?
All the graphs represent similar parabolas because all parabolas are similar. The graphs of theparabolas described in parts (i) and (ii) are congruent to the original parabola because the value of๐๐ is the same.
I notice the distance between the focus and directrix is the same as in the graph of ๐๐ and the vertex is (0, 0). Since the directrix is above the focus, this transformation is a reflection across the ๐ฅ๐ฅ-axis.
I notice the distance between the focus and directrix is 8 times the corresponding distance on the graph of ๐๐. I also notice the directrix is vertical and to the left of the focus, which means that the graph has been rotated 90ยฐ clockwise.
This parabola is a vertical scaling of the graph of ๐๐ by a factor of ๐๐
๐๐ and a
clockwise rotation of the resulting image by ๐๐๐๐ยฐ about the origin.
This parabola is a reflection of thegraph of ๐๐ across the ๐๐-axis.
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Lesson 35: Are All Parabolas Similar?
Since I performed rigid transformations on the graph of ๐ฆ๐ฆ = 2๐ฅ๐ฅ2, the transformed graph is congruent to it.
3. Write the equation of a parabola congruent to ๐ฆ๐ฆ = 2๐ฅ๐ฅ2 that contains the point (1,โ4). Describe thetransformations that would take this parabola to your new parabola.
Note: There are many possible correct answers.
๐๐ = ๐๐(๐๐ โ ๐๐)๐๐ โ ๐๐
The graph of the equation ๐๐ = ๐๐๐๐๐๐ has its vertex at theorigin. The graph of the equation ๐๐ = ๐๐(๐๐ โ ๐๐)๐๐ โ ๐๐ is atranslation ๐๐ unit to the right and ๐๐ units down, so its vertexis (๐๐,โ๐๐).
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Lesson 36: Overcoming a Third Obstacle to FactoringโWhat If There Are No Real Number Solutions?
The graph of the first equation is a parabola with vertex at (0, 1) that opens down. The graph of the second equation is a line with slope โ1 and ๐ฆ๐ฆ-intercept 2.
I know that given ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, if ๐๐2 โ 4๐๐๐๐ < 0, there are no real solutions to the equation. Since ๐ฅ๐ฅ represents the ๐ฅ๐ฅ-coordinate of the point of intersection of the graphs, there are no points of intersection.
Lesson 36: Overcoming a Third Obstacle to FactoringโWhat If
There Are No Real Number Solutions?
1. Solve the following systems of equations, or show that no real solution exists. Graphically confirm youranswers.a. 3๐ฅ๐ฅ โ 2๐ฆ๐ฆ = 10
6๐ฅ๐ฅ + 4๐ฆ๐ฆ = 16
So, ๐๐ = โ๐๐๐๐. Then ๐๐๐๐ โ ๐๐๏ฟฝโ๐๐
๐๐๏ฟฝ = ๐๐๐๐ so ๐๐๐๐ + ๐๐ = ๐๐๐๐ and
thus ๐๐ = ๐๐.
The solution is ๏ฟฝ๐๐,โ๐๐๐๐๏ฟฝ.
b. 3๐ฅ๐ฅ2 + ๐ฆ๐ฆ = 1๐ฅ๐ฅ + ๐ฆ๐ฆ = 2
๐๐ = โ๐๐+ ๐๐
Substituting โ๐๐ + ๐๐ for ๐๐gives๐๐๐๐๐๐ + (โ๐๐ + ๐๐) = ๐๐, sothe discriminant is
๐๐๐๐ โ ๐๐๐๐๐๐ = (โ๐๐)๐๐ โ ๐๐(๐๐)(๐๐) = โ๐๐๐๐.
Since the discriminant is negative, there are no real solutions to the equation ๐๐๐๐๐๐ โ ๐๐ + ๐๐ = ๐๐.
There are no real solutions to the system.
I can solve the system using elimination like we did in Lessons 30โ31.
โ๐๐(๐๐๐๐ โ ๐๐๐๐) = โ๐๐(๐๐๐๐) ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐
โ๐๐๐๐+ ๐๐๐๐ = โ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐
๐๐๐๐ = โ๐๐
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Lesson 36: Overcoming a Third Obstacle to FactoringโWhat If There Are No Real Number Solutions?
I know when a linear equation is written in the form ๐ฆ๐ฆ = ๐๐๐ฅ๐ฅ + ๐๐, ๐๐ represents the slope of the line.
I can complete the square to help me identify key features of the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 7.
A translation upward by ๐๐ units will result in aparabola that opens up with a vertex at (๐๐,๐๐).Since the vertex is above the ๐๐-axis and theparabola opens up, the graph will not intersect the๐๐-axis.
2. Find the value of ๐๐ so that the graph of the following system of equations has no solution.2๐ฅ๐ฅ โ ๐ฆ๐ฆ + 8 = 0 ๐๐๐ฅ๐ฅ + 3๐ฆ๐ฆ โ 9 = 0
If ๐๐๐๐ โ ๐๐ + ๐๐ = ๐๐, then ๐๐ = ๐๐๐๐ โ ๐๐.
If ๐๐๐๐ + ๐๐๐๐ โ ๐๐ = ๐๐, then ๐๐ = โ๐๐๐๐๐๐ + ๐๐.
Since there are no solutions to the system, the lines are parallel and the slopes of the lines must be equal. Therefore, ๐๐ = โ๐๐
๐๐, and it follows that ๐๐ = โ๐๐.
3. Consider the equation ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 7 = 0.a. Offer a geometric explanation to why the equation has two real solutions.
๐๐๐๐ โ ๐๐๐๐ + ๐๐ = ๐๐ (๐๐๐๐ โ ๐๐๐๐ + ๐๐) + ๐๐ โ ๐๐ = ๐๐
(๐๐ โ ๐๐)๐๐ โ ๐๐ = ๐๐
The graph of this equation is a parabola that opens upward with vertex (๐๐,โ๐๐). Since the graph opens up and has a vertex below the ๐๐-axis, the graph will intersect the ๐๐-axis in two locations, which means the equation has two real solutions.
b. Describe a geometric transformation to the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 7 so the graph intersects the๐ฅ๐ฅ-axis in one location.
A translation upward by ๐๐ units will result in a graph that opens up with a vertex at (๐๐,๐๐). Sincethe vertex is on the ๐๐-axis and the parabola opens up, it will intersect the ๐๐-axis in only onelocation.
c. Describe a geometric transformation to the graphof ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 7 so the graph does notintersect the ๐ฅ๐ฅ-axis.
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Lesson 36: Overcoming a Third Obstacle to FactoringโWhat If There Are No Real Number Solutions?
d. Find the equation of a line that would intersect the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 7 in one location.
The graph of ๐๐ = โ๐๐ is a horizontal line tangent to the vertex of the graph of ๐๐ = ๐๐๐๐ โ ๐๐๐๐ + ๐๐.
Note: There are multiple correct solutions to parts (b)โ(d).
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Lesson 37: A Surprising Boost from Geometry
Lesson 37: A Surprising Boost from Geometry
1. Express the quantities below in ๐๐ + ๐๐๐๐ form, and graph the corresponding points on the complex plane.Label each point appropriately.a. (12 + 2๐๐) โ 3(6 โ ๐๐)
๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = โ๐๐ + ๐๐๐๐
b. (4 + ๐๐)(5 + 3๐๐)
(๐๐ + ๐๐)(๐๐ + ๐๐๐๐) = ๐๐(๐๐) + ๐๐(๐๐๐๐) + ๐๐(๐๐) + ๐๐(๐๐๐๐)= ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ โ ๐๐ = ๐๐๐๐ + ๐๐๐๐๐๐
c. ๐๐(4 โ ๐๐)(2 + 3๐๐)
2. Find the real value of ๐ฅ๐ฅ and ๐ฆ๐ฆ in each of the following equations using the fact that if ๐๐ + ๐๐๐๐ = ๐๐ + ๐๐๐๐,then ๐๐ = ๐๐ and ๐๐ = ๐๐.
2(4 + ๐ฅ๐ฅ) โ 5(3๐ฆ๐ฆ + 2)๐๐ = 4๐ฅ๐ฅ โ (3 + ๐ฆ๐ฆ)๐๐
Since ๐๐ = ๐๐, it follows that ๐๐(๐๐ + ๐๐) = ๐๐๐๐. Then ๐๐ + ๐๐๐๐ = ๐๐๐๐, so ๐๐๐๐ = ๐๐, and ๐๐ = ๐๐.
Since ๐๐ = ๐ ๐ , it follows that โ๐๐(๐๐๐๐ + ๐๐) = โ(๐๐+ ๐๐). Then โ๐๐๐๐๐๐ โ ๐๐๐๐ = โ๐๐ โ ๐๐, so ๐๐๐๐๐๐ = โ๐๐, and ๐๐ = โ๐๐
๐๐.
I know ๐๐2 = โ1. ๐๐๏ฟฝ๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐๐๐๏ฟฝ = ๐๐๏ฟฝ๐๐ + ๐๐๐๐๐๐ โ ๐๐(โ๐๐)๏ฟฝ
= ๐๐(๐๐๐๐ + ๐๐๐๐๐๐) = ๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐ = โ๐๐๐๐ + ๐๐๐๐๐๐
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Lesson 37: A Surprising Boost from Geometry
3. Express each of the following complex numbers in ๐๐ + ๐๐๐๐ form.a. ๐๐23
๐๐๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐๐ = ๏ฟฝ๐๐๐๐๏ฟฝ๐๐ โ ๐๐๐๐ = ๐๐ โ ๐๐๐๐ = ๐๐๐๐ โ ๐๐ = โ๐๐
b. (1 + ๐๐)5
(๐๐ + ๐๐)๐๐ = (๐๐ + ๐๐)๐๐(๐๐ + ๐๐)= ๏ฟฝ(๐๐ + ๐๐)๐๐๏ฟฝ๐๐ โ (๐๐ + ๐๐)
= ๏ฟฝ๐๐ + ๐๐๐๐ + ๐๐๐๐๏ฟฝ๐๐(๐๐ + ๐๐)= (๐๐ + ๐๐๐๐ โ ๐๐)๐๐(๐๐ + ๐๐) = (๐๐๐๐)๐๐(๐๐ + ๐๐) = ๐๐๐๐๐๐(๐๐ + ๐๐) = โ๐๐(๐๐ + ๐๐) = โ๐๐ โ ๐๐๐๐
c. โโ4 โ โโ25
โโ๐๐ โ โโ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐๐๐๐๐ = โ๐๐๐๐
I know ๐๐4 = ๐๐2 โ ๐๐2 = โ1 โ โ1 = 1.
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Lesson 38: Complex Numbers as Solutions to Equations
Lesson 38: Complex Numbers as Solutions to Equations
1. Solve the equation 5๐ฅ๐ฅ2 + 3๐ฅ๐ฅ + 1 = 0.
This is a quadratic equation with ๐๐ = ๐๐, ๐๐ = ๐๐, and ๐๐ = ๐๐.
๐๐ =โ(๐๐) ยฑ ๏ฟฝ๐๐๐๐ โ ๐๐(๐๐)(๐๐)
๐๐(๐๐) =โ๐๐ ยฑ โโ๐๐๐๐
๐๐๐๐=โ๐๐ ยฑ ๐๐โ๐๐๐๐
๐๐๐๐
Thus, the solutions are โ ๐๐๐๐๐๐
+ ๐๐โ๐๐๐๐๐๐๐๐
and โ ๐๐๐๐๐๐โ ๐๐โ๐๐๐๐
๐๐๐๐.
2. Suppose we have a quadratic equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 so that ๐๐ โ 0, ๐๐ = 0, and ๐๐ < 0. Does thisequation have one solution or two distinct solutions? Are the solutions real or complex? Explain how youknow.
Because ๐๐ = ๐๐, ๐๐๐๐๐๐ + ๐๐ = ๐๐, so ๐๐๐๐๐๐ = โ๐๐.
Then since ๐๐ < ๐๐, we know โ๐๐ > ๐๐, and it must be that ๐๐๐๐๐๐ > ๐๐.
If ๐๐ > ๐๐, then ๐๐๐๐ > ๐๐, and the equation has two distinct real solutions.
Otherwise, if ๐๐ < ๐๐, then ๐๐๐๐ < ๐๐, and there are two distinct complex solutions.
3. Write a quadratic equation in standard form such that 2๐๐ and โ2๐๐ are its solutions.(๐๐ โ ๐๐๐๐)(๐๐ + ๐๐๐๐) = ๐๐
๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐ ๐๐๐๐ + ๐๐ = ๐๐
4. Is it possible that the quadratic equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 has two real solutions if ๐๐ and ๐๐ areopposites?
I can use the multiplication property of inequality to isolate ๐ฅ๐ฅ2.
I know if ๐๐ is a solution to the polynomial equation ๐๐(๐ฅ๐ฅ) = 0, then (๐ฅ๐ฅ โ ๐๐) is a factor of ๐๐.
I know that the equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 has two distinct real solutions when ๐๐2 โ 4๐๐๐๐ > 0.
I can use the quadratic formula to solve the equation.
Since ๐๐ and ๐๐ are opposites, ๐๐ = โ๐๐. Then๐๐๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐ โ ๐๐(๐๐)(โ๐๐) = ๐๐๐๐ + ๐๐๐๐๐๐.
Since ๐๐๐๐ is nonnegative and ๐๐๐๐๐๐ is positive, ๐๐๐๐ + ๐๐๐๐๐๐ > ๐๐.
Therefore, there are always two real solutions to the equation ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐ = ๐๐ when ๐๐ = โ๐๐.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 38: Complex Numbers as Solutions to Equations
5. Let ๐๐ be a real number, and consider the quadratic equation ๐๐๐ฅ๐ฅ2 + (๐๐ โ 2)๐ฅ๐ฅ + 4 = 0.a. Show that the discriminant of ๐๐๐ฅ๐ฅ2 + (๐๐ โ 2)๐ฅ๐ฅ + 4 = 0 defines a quadratic function of ๐๐.
Here, ๐๐ = ๐๐, ๐๐ = ๐๐ โ ๐๐, and ๐๐ = ๐๐.
๐๐๐๐ โ ๐๐๐๐๐๐ = (๐๐ โ ๐๐)๐๐ โ ๐๐ โ (๐๐) โ ๐๐= ๐๐๐๐ โ ๐๐๐๐ + ๐๐ โ ๐๐๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐
With ๐๐ unknown, we can write ๐๐(๐๐) = ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐, which is a quadratic function of ๐๐.
b. Find the zeros of the function in part (a), and make a sketch of itsgraph.
If ๐๐(๐๐) = ๐๐, then ๐๐ = ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐.
๐๐ =โ(โ๐๐๐๐) ยฑ ๏ฟฝ(โ๐๐๐๐)๐๐ โ ๐๐(๐๐)(๐๐)
๐๐
=๐๐๐๐ ยฑ โ๐๐๐๐๐๐
๐๐= ๐๐๐๐ ยฑ ๐๐โ๐๐
The zeros of the function are ๐๐๐๐ + ๐๐โ๐๐ and ๐๐๐๐ โ ๐๐โ๐๐.
c. For what value of ๐๐ are there two complex solutions to thegiven quadratic equation?
There are two complex solutions when ๐๐(๐๐) < ๐๐.
This occurs for all real numbers ๐๐ such that ๐๐๐๐ โ ๐๐โ๐๐ < ๐๐ < ๐๐๐๐ + ๐๐โ๐๐.
I know the discriminant of a quadratic equation written in the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is ๐๐2 โ 4๐๐๐๐.
I know the discriminant is less than zero where the graph of ๐๐ is below the ๐ฅ๐ฅ-axis, which is between the values of the zeros.
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81
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 39: Factoring Extended to the Complex Realm
Lesson 39: Factoring Extended to the Complex Realm
1. Rewrite each expression in standard form.a. (๐ฅ๐ฅ โ 4๐๐)(๐ฅ๐ฅ + 4๐๐)
(๐๐ โ ๐๐๐๐)(๐๐ + ๐๐๐๐) = ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ + ๐๐๐๐
b. (๐ฅ๐ฅ + 2 โ ๐๐)(๐ฅ๐ฅ โ 2 + ๐๐)
3. Write a polynomial equation of degree 4 in standard form that has the solutions 2๐๐, โ2๐๐, 3, โ2.
(๐๐ โ ๐๐๐๐)(๐๐ + ๐๐๐๐)(๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐ ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ = ๐๐
๐๐๐๐๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ + ๐๐๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ = ๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ = ๐๐
๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐ โ ๐๐๐๐ = ๐๐
4. Find the solutions to ๐ฅ๐ฅ4 โ 8๐ฅ๐ฅ2 โ 9 = 0 and the ๐ฅ๐ฅ-intercepts of the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ4 โ 8๐ฅ๐ฅ2 โ 9.๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐
(๐๐ + ๐๐)(๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐
The solutions are โ๐๐, ๐๐, โ๐๐, and ๐๐.
The ๐๐-intercepts are โ๐๐ and ๐๐.
5. Explain how you know that the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ4 + 13๐ฅ๐ฅ2 + 36 has no ๐ฅ๐ฅ-intercepts.
If ๐๐๐๐ + ๐๐๐๐๐๐๐๐ + ๐๐๐๐ = ๐๐, then ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ = ๐๐, so (๐๐ + ๐๐๐๐)(๐๐ โ ๐๐๐๐)(๐๐ + ๐๐๐๐)(๐๐ โ ๐๐๐๐) = ๐๐.
The solutions to the quadratic equation are โ๐๐๐๐, ๐๐๐๐, โ๐๐๐๐, and ๐๐๐๐. Since none of the solutions are real,the graph has no ๐๐-intercepts.
I know if ๐๐ is a solution to ๐๐(๐ฅ๐ฅ) = 0, then (๐ฅ๐ฅ โ ๐๐) is a factor, so the factors must be (๐ฅ๐ฅ โ 2๐๐), (๐ฅ๐ฅ +2๐๐), (๐ฅ๐ฅ โ 3), and (๐ฅ๐ฅ + 2).
I recognize this pattern as the factoring for the sum of squares: (๐๐ + ๐๐๐๐)(๐๐ โ ๐๐๐๐) = ๐๐2 + ๐๐2.
I recognize this pattern as the factoring for the difference of two squares:
(๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐2 โ ๐๐2. I can see that ๐๐ = ๐ฅ๐ฅ and ๐๐ = (2 โ ๐๐).
I can also let ๐ข๐ข = ๐ฅ๐ฅ2 and factor the equation ๐ข๐ข2 โ 8๐ข๐ข โ 9.
I know only real solutions to the equation are ๐ฅ๐ฅ-intercepts of the graph.
(๐๐ + ๐๐ โ ๐๐)(๐๐ โ ๐๐ + ๐๐) = ๏ฟฝ๐๐ + (๐๐ โ ๐๐)๏ฟฝ๏ฟฝ๐๐ โ (๐๐ โ ๐๐)๏ฟฝ = ๐๐๐๐ โ (๐๐ โ ๐๐)๐๐ = ๐๐๐๐ โ ๏ฟฝ๐๐ โ ๐๐๐๐ + ๐๐๐๐๏ฟฝ = ๐๐๐๐ โ (๐๐ โ ๐๐๐๐) = ๐๐๐๐ โ ๐๐ + ๐๐๐๐
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82
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 40: Obstacles ResolvedโA Surprising Result
Lesson 40: Obstacles ResolvedโA Surprising Result
1. Write each quadratic function below as a product of of linear factors.a. ๐๐(๐ฅ๐ฅ) = 9๐ฅ๐ฅ2 + 16
๐๐(๐๐) = (๐๐๐๐ + ๐๐๐๐)(๐๐๐๐ โ ๐๐๐๐)
b. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 12
๐๐(๐๐) = ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐ โ ๐๐๐๐ + ๐๐ + ๐๐
๐๐(๐๐) = (๐๐๐๐ โ ๐๐๐๐ + ๐๐) โ (โ๐๐) = (๐๐ โ ๐๐)๐๐ โ (๐๐โ๐๐)๐๐
๐๐(๐๐) = ๏ฟฝ(๐๐ โ ๐๐) โ ๐๐โ๐๐๏ฟฝ ๏ฟฝ(๐๐ โ ๐๐) + ๐๐โ๐๐๏ฟฝ
c. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + 10๐ฅ๐ฅ
๐๐(๐๐) = ๐๐๏ฟฝ๐๐๐๐ + ๐๐๐๐๏ฟฝ = ๐๐(๐๐ + ๐๐โ๐๐๐๐)(๐๐ โ ๐๐โ๐๐๐๐)
2. For each cubic function below, one of the zeros is given. Express each function as a product of linearfactors.a. ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ3 + 3๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ + 3; ๐๐(1) = 0
b. ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 1; ๐๐(โ1) = 0
๐๐(๐๐) = ๐๐๐๐(๐๐ + ๐๐) + ๐๐(๐๐ + ๐๐) = ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ(๐๐ + ๐๐) = (๐๐ โ ๐๐)(๐๐ + ๐๐)(๐๐ + ๐๐)
3. Consider the polynomial function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ6 โ 65๐ฅ๐ฅ3 + 64.a. How many linear factors does ๐ฅ๐ฅ6 โ 65๐ฅ๐ฅ3 + 64 have? Explain.
There are ๐๐ linear factors because the fundamental theorem of algebra states that a polynomialwith degree ๐๐ can be written as a product of ๐๐ linear factors.
I can factor this expression using the sum of squares identity, which we discovered in Lesson 39.
If I complete the square, I can rewrite this quadratic expression as the difference of two squares.
Since ๐๐2 = โ3, I know ๐๐ = ยฑโโ3 = ยฑ๐๐โ3.
I can find the quotient of ๐๐ divided by (๐ฅ๐ฅ โ 1) using the long division algorithm from Lesson 5 or the reverse tabular method from Lesson 4.
The factor theorem tells me that if ๐๐(1) = 0, then (๐ฅ๐ฅ โ 1) is a factor of ๐๐.
I can factor this polynomial expression by grouping.
๐๐(๐๐) = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐๐๐ + ๐๐๐๐ โ ๐๐๏ฟฝ= (๐๐ โ ๐๐)(๐๐๐๐ โ ๐๐)(๐๐ + ๐๐)
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83
Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015
2015-16 M1 ALGEBRA II
Lesson 40: Obstacles ResolvedโA Surprising Result
b. Find the zeros of ๐๐.
๐๐(๐๐) = ๐๐๐๐ โ ๐๐๐๐๐๐๐๐ + ๐๐๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐ โ ๐๐๏ฟฝ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐๏ฟฝ(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ
To find the zeros of ๐๐๐๐ + ๐๐๐๐ + ๐๐:
๐๐ =โ๐๐ยฑ โ๐๐๐๐ โ ๐๐๐๐๐๐
๐๐๐๐=โ๐๐ ยฑ โ๐๐๐๐ โ ๐๐๐๐
๐๐=โ๐๐ ยฑ โโ๐๐๐๐
๐๐=โ๐๐ ยฑ โ๐๐๐๐โโ๐๐
๐๐= โ๐๐ยฑ ๐๐๐๐โ๐๐
To find the zeros of ๐๐๐๐ + ๐๐ + ๐๐:
๐๐ =โ๐๐ยฑ โ๐๐๐๐ โ ๐๐๐๐๐๐
๐๐๐๐=โ๐๐ยฑ โ๐๐ โ ๐๐
๐๐=โ๐๐ยฑ โโ๐๐
๐๐=โ๐๐ ยฑ ๐๐โ๐๐
๐๐= โ
๐๐๐๐
ยฑโ๐๐๐๐๐๐.
The zeros of ๐๐ are ๐๐, โ๐๐ + ๐๐๐๐โ๐๐, โ๐๐ โ ๐๐๐๐โ๐๐, ๐๐, โ๐๐๐๐
+ โ๐๐๐๐๐๐, โ๐๐
๐๐โ โ๐๐
๐๐๐๐.
4. Consider the polynomial function ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ4 + ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 โ 9๐ฅ๐ฅ โ 10.a. Use the graph to find the real zeros of ๐๐.
The real zeros are โ๐๐ and ๐๐.
b. Confirm that the zeros are correct by evaluating the function๐๐ at those values.
๐ท๐ท(โ๐๐) = (โ๐๐)๐๐ + (โ๐๐)๐๐ + (โ๐๐)๐๐ โ ๐๐(โ๐๐)โ ๐๐๐๐ = ๐๐ ๐ท๐ท(๐๐) = (๐๐)๐๐ + (๐๐)๐๐ + (๐๐)๐๐ โ ๐๐(๐๐) โ ๐๐๐๐ = ๐๐
c. Express ๐๐ in terms of linear factors.
I can factor each of these expressions using the difference of cubes pattern: ๐๐3 โ ๐๐3 = (๐๐ โ ๐๐)(๐๐2 + ๐๐๐๐ + ๐๐2).
๐ท๐ท(๐๐) = (๐๐ + ๐๐)(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๏ฟฝ= (๐๐ + ๐๐)(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐ + ๐๐๏ฟฝ = (๐๐ + ๐๐)(๐๐ โ ๐๐)๏ฟฝ(๐๐ + ๐๐)๐๐ โ (๐๐๐๐)๐๐๏ฟฝ = (๐๐ + ๐๐)(๐๐ โ ๐๐)๏ฟฝ(๐๐ โ ๐๐) + ๐๐๐๐๏ฟฝ๏ฟฝ(๐๐ โ ๐๐) โ ๐๐๐๐๏ฟฝ
The zero product property tells me that I can set each factor equal to 0 to find the zeros of the polynomial.
I can use the reverse tabular method or the long division algorithm to factor (๐ฅ๐ฅ + 1)(๐ฅ๐ฅ โ 2) from ๐๐.
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Homework Helper A Story of Functions
ALG II-M1-HWH-1.3.0-08.2015