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Chapter 0: Algebra II Review
Topic 1: Simplifying Polynomials & Exponential Expressions – p. 2
- Homework: Worksheet
Topic 2: Radical Expressions – p. 32
- Homework: p. 45 #33-74 Even
Topic 3: Factoring All Ways – p. 58
- Homework: p 68 & 69 #1-92 Even
Topic 4: Rational Expressions – p. 72
- Homework: Worksheet
Topic 5: Complex Fractions – p. 80
- Homework: Worksheet
Topic 6: Completing the Square – p. 98
- Homework: p. 106 #91-110 (Complete the Square)
2 | P a g e
Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 1: Simplifying Polynomials and Exponential Expressions
Polynomial Operations:
Addition/Subtraction: Combine like-terms only
1. 2.
3. 4. Subtract from
Multiplication: Every term by every term
1. 2.
3. 4.
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Division:
1.
2.
Exponent Rules:
Remember: Exponents are always a little off from regular arithmetic rules.
Addition/Subtraction: Combine coefficients of
like-terms; exponents are unchanged
Multiplication: Multiply coefficients; add
exponents of like-bases
Division: Divide coefficients; subtract
exponents of like-bases
Negative Exponents: “I’m stuck on the wrong
side of the fraction line!” Hint: deal with these
first in complex questions!
Fractional Exponents: Power over Root
1. 2. 3.
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4. 5. 6.
7.
8. 9.
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Homework:
Perform the indicated operation.
1. (9y2 - 12y + 5) - (12y2 + 6y - 11)
2. 8(7r + y) - 3(5r - 2)
3. 2(y2 + 4y) + 6y(y - 3) 4. (8r -1) - 3(10r - 8)
5. (3g3 - 2g2 + 1)(g - 4) 6. (9 - y2)(2y + 1)
7.
8.
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Use your knowledge of exponent rules to simplify the following expressions
9. 10.
11. 12.
13. 14.
15.
16.
17.
18.
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19. 20.
21. 22.
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Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 2: Radical Expressions
Do Now:
Simplify the following radical expressions
1. 2.
3. 4.
Adding & Subtracting Radicals:
Just like anything else, we can only _____________________________________________________________
When adding or subtracting radicals, both the ______________________ AND the
____________________________ must be exactly the same
Before we begin to combine, we must first _________________________________________________.
Example:
Add:
1. Simplify each of the terms
2. Combine the like terms (add/subtract the
coefficients of the like-radicands)
1. 2.
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Multiplying Radicals: Multiply the numbers outside the radicals… the ________________________
Multiply the numbers inside the radicals… the _______________________
Simplify the radicals in your final answers. Do not simplify until __________________________!!!
Example:
Multiply:
1. Multiply coefficients; Multiply Radicands
2. Simplify at the end
*observe: if we simplified at the beginning,
we’d have to simplify again at the end!
1.
2.
3.
4.
3. 4.
10 | P a g e
5.
6.
7.
8.
9.
10.
Pairs of binomials like #10 are called:
Definition: Conjugate Pairs -
_____________________________________________________________________________________
The result of multiplying conjugate pairs of radical expressions will ALWAYS be an INTEGER.
When multiplying conjugate pairs, we can skip FOIL and just multiple first & last terms. Be VERY sure you are
dealing with conjugate pairs before you take this shortcut!
Example:
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Dividing Radicals: Divide the numbers outside the radicals, divide the numbers inside the radicals.
Simplify the radicals in your final answers. If necessary, _____________________ ________
____________________
Example: Rationalizing Monomial Denominators
Divide:
1. Divide as much as possible
2. Simplify at the end – Rationalize if necessary
Example: Rationalizing Binomial Denominators
Divide:
1. Divide as much as possible (usually nothing is possible with
binomial denominators)
2. Simplify at the end – Rationalize if necessary
1.
2.
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3.
4.
5.
6.
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Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 3: Factoring ALL Ways
Factoring:
When we factor it is important to remember that ____________________________________________
_______________________________; we are simply rewriting it in an equivalent form.
GCF or “Greatest Common Factor” (review!):
Factoring by GCF means that we “__________________” what the terms have in _______________. This can be a
combination of numbers, variables, or both.
Factor out by GCF.
1. 2. –
3. – 4. –
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“DOTS” Factoring (review!):
Another type of factoring stems from _______________________ multiplication. This type of factoring is known as
Difference of Two Squares or “DOTS” factoring. When we factor these types of expressions, we _____________
conjugate multiplication.
Write each of the following binomials as the product of a conjugate pair.
1. – 2. –
3.
4.
Trinomial Case I Factoring (review!):
Another type of factoring is trinomial factoring. This is when we have a trinomial. In Case I factoring, the leading
coefficient is _________.
To factor these, it is helpful to look at the ______________ term and the ___________________ term.
Write each of the following trinomials in factored form.
1. 2.
3. 4.
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Factor By Grouping (review!)
We can factor by grouping when we have a polynomial that has __________________________________________.
Class Example #1: Only one variable
Steps:
1. Put terms in descending order, or with other like-factor
terms
2. Group terms in sets of 2
3. Factor each group
4. Rearrange
Use FOIL to check!
Class Example #2: More than one variable
Steps:
1. Put terms in descending order, or with other like-factor
terms
2. Group terms in sets of 2
3. Factor each group
4. Rearrange
Use FOIL to check
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Grouping Examples: 1. 2.
3. 4.
Factor Trinomials - with a leading coefficient (review!)
“Case II Factoring” To solve these examples, we use factoring by grouping by "splitting" up the middle into two factors.
Always make sure that every example is written in standard form before you try to split & factor.
Class Example #3
Steps:
1. Decide on factors & signs
2. Rewrite as 4 terms
3. Factor by grouping
4. Rearrange
Use FOIL to check!
Case II Factoring Examples: 1. 2. 5
3. 4.
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Factoring Sums & Differences of Two Cubes
General Rules
be very aware of what signs are used & when!
Worth studying and committing to memory. This will be used this year and even more in Calculus courses.
Class Example #6:
Steps:
1. Identify the cube roots of both terms
2. Plug in to the appropriate pattern (given above)
Distribute carefully to check!
Sum & Difference of Two Cubes Examples:
1. 2.
3. 4.
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Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 4: Rational Expressions
Define:
Rational Expression:
Domain:
Limits to domain in the real number system
Recall that the set of values which make up your domain is typically unlimited. You can plug any number you want in to
an expression or a function. (The output depends on the rule given and can vary greatly and is only limited by the rule of
the expression or function)
There is one large notable, very important, exception to the domain of rational expressions.
A denominator of a fraction can never ____________________________________!!!!!!
Examples: Determine the numbers that must be excluded from each domain:
1)
2)
3)
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Simplifying Rational Expressions
We must ensure we understand the structure of the expression we are simplifying. It is not proper to split up terms that
are joined by addition or subtraction. Rather, we can only cancel factors that appear in both the numerator and
denominator
Examples: Simplify completely by ‘factor & cancel’:
1)
2)
Review of Rational Expression Operations:
Multiplication & Division
To multiply rational expressions, factor all numerators and denominators completely. Then, across all numerator factors
and all denominator factors, cancel any matches. Multiply what remains and ensure your final answer is in simplest
form.
To divide, FIRST, perform ‘keep change reciprocal’, then proceed the same way.
Examples: Perform the indicated operation and simplify:
1)
2)
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3)
4)
Addition & Subtraction
Just like any type of adding or subtracting, we can only combine like terms. In a rational expression, like terms happen
when we have COMMON DENOMINATORS. Therefore, we must get common denominators before we can combine.
Review with arithmetic:
Least common denominator:
Find the LCD Multiply by what’s missing to get common denominators Add across
Practice the same method with a rational expression:
Steps:
1) Change subtraction if necessary.
2) Find the LCD.
3) Multiply each term by what is missing.
4) Add across.
5) Simplify if possible.
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Examples: Perform the indicated operation and simplify:
1)
2)
3)
4)
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Homework:
Simplify each rational expression:
1.
2.
3.
4.
Indicate the values for which the rational expression is undefined:
5.
6.
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Perform the indicated operation and simplify, if possible:
7.
8.
9.
10.
11.
12.
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13.
14.
15.
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Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 5: Complex Fractions
Simplifying Complex Fractions
Two methods for solving complex fractions are presented below. Each has a preferred time to be used, but both can be
used in any situation with careful setup. Try both options to ensure you’re fluent in both.
Class Example #1: Single rational expressions in the numerator & denominator
Rewrite left to right. Keep, change, reciprocal. Simplify.
Class Example #2: Multiple rational expressions in the numerator and/or denominator
STEPS:
1. Clean it up: Ensure ALL terms are fractions.
2. Find the least common denominator (LCD) of all of the fractions.
We do this by factoring all of the denominators of the smaller
fractions.
3. Multiply all of the fractions by the LCD we found in step 2. (By
doing this, we are multiplying the numerator & denominator of
our complex fraction by the LCD). ALL OF YOUR LITTLE
DENOMINATORS MUST CANCEL!
4. Simplify whenever possible.
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Complex Fraction Examples:
1)
2)
3)
4)
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Complex Fractions Containing Radicals
In order to simplify complex fractions with radicals, we must be able to multiply expressions with radicals
There are 2 Cases:
1. When the expression under the radical is different from the expression without the radical:
--> Here we cannot simplify further, so the expression is left as
2. When the expressions are the same inside and outside the radical:
--> Here we will follow the following steps:
1. Rewrite the radical with exponents (no radical)
2. To multiply, we will add the exponents
3. Rewrite as a radical
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Examples:
1. 2. 3.
4. 5. 6.
Complex Fractions with Radicals
Simplify each of the following.
1.
2.
3.
4.
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Homework:
Express each complex fraction or rational expression in simplest form:
1.
2.
3.
4.
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Simplify each of the following.
5.
6.
7.
8.
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Name: _____________________________________________ Date: ____________________ Period: _________
Chapter 0: Algebra II Review
Topic 6: Completing the Square
Completing the Square: We will force the left-side of the equation to become a perfect square trinomial.
Completing the Square 1. Move the constant term to the other side.
2. Be sure the coefficient of the highest power is one. If it is not, factor out the coefficient from
3. Create a perfect square trinomial by adding
(to both sides!) Be careful if there was a constant factored out.
4. Factor the perfect square; add the constants together.
5. Isolate the variable to solve. (Square root both sides, remove what remains)
Examples:
1. 2.
3. 4.
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5.
6.
7.