Embed Size (px)
Citation preview
NC Math 3
Modelling with
Polynomials
2
Introduction to Polynomials; Polynomial Graphs and Key Features
Polynomial Vocabulary Review Expression:
Equation:
Terms:
o Monomial, Binomial, Trinomial, Polynomial
Degree:
o Constant, Linear, Quadratic, Cubic, Quartic
POLYNOMIAL
NUMBER OF TERMS
CLASSIFICATION BY
TERMS
DEGREE
CLASSFICATION BY
DEGREE
f(x) = 5
g(x) = 4x – 3
p(x) = –2x5
w(x) = x4 – 4x + 2
y = – 4x2 + x + 9
h(x) = 4x3 + x2 – 9x + 2
https://student.desmos.com/?prepopulateCode=vcf7p
3
The number “k” is said to be a zero of a polynomial if f(k) = 0. “k” is often referred to as the root or solution If “k” is a real number, then f(k) = 0 means that the graph crosses the x-axis at that value.
“k” can also be referred to as an x-intercept Check out the graphs below and identify any values that represent a zero/solution/root.
A. B. factored equation: _____________________ factored equation: _____________________
C. D. factored equation: _____________________ factored equation: _____________________
4
WATCH OUT! Multiplicities of Zeros If c is a zero of the function P and the corresponding factor (x − c) occurs exactly m times in the factorization of P then we say that c is a zero of multiplicity m. One can show that the graph of P crosses the x-axis at c if the multiplicity m is odd and does not cross the x-axis if m is even.
5
Polynomial Degrees, Roots, Factored Form and Turning Points
Degree
Number of Unique Real
Roots (indicate if a root has a
multiplicity greater than 1)
Number of Non-Real
Roots Sketch a Graph
Possible Equation of the graph in Factored Form (Only if all roots
are real)
Number of Turning Points (maxima and
minima)
1 1 0
2 2
2 2
1 (multiplicity of 2)
0
6
Degree
Number of Unique Real
Roots (indicate if a root has a
multiplicity greater than 1)
Number of Non-Real
Roots Sketch a Graph
Possible Equation of the graph in Factored Form (Only if all roots
are real)
Number of Turning Points (maxima and
minima)
1 2
3 3
3 1 (with a
multiplicity of 3)
3 1 (multiplicity of
2) 1
7
Degree
Number of Unique Real
Roots (indicate if a root has a
multiplicity greater than 1)
Number of Non-Real
Roots Sketch a Graph
Possible Equation of the graph in Factored Form (Only if all roots
are real)
Number of Turning Points (maxima and
minima)
2 2
4 4
4 4
4 1 (multiplicity of
2)
8
Degree
Number of Unique Real
Roots (indicate if a root has a
multiplicity greater than 1)
Number of Non-Real
Roots Sketch a Graph
Possible Equation of the graph in Factored Form (Only if all roots
are real)
Number of Turning Points (maxima and
minima)
2 (each having a multiplicity of
2) 0
4 (1 having a multiplicity of 2, and the other
two are unique)
0
Summary Questions:
1. What is the relationship between degree and number of roots?
2. What is the relationship between degree and number of turning points?
3. What is the relationship between the factored form of the equation and the x-intercepts?
4. Why do you think that non-real roots always come in pairs? (**Hint** Think about the quadratic formula, why can you not get only one non-real answer when you use the quadratic formula?)
9
The high and low points on a graph are called the extrema of the function. An extremum that is higher or lower than any other points nearby is called a relative extremum.
A relative extremum (the plural of extremum is extrema) that is higher than points nearby is called relative maximum. A relative extremum that is lower than points nearby is called a relative minimum. A function’s absolute extremum occurs at the highest or lowest point on a function. The highest point on a function is called the absolute maximum and the lowest point on a function is called the absolute minimum.
Can you identify these in the graphs on the previous page? The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. For any polynomial, the end behavior is determined by the term that contains the highest power of x, because when x is large, the other terms are relatively insignificant in size.
Given the graph of the polynomial below - State the intervals where the graph is
increasing/decreasing (think “slope”)
- State the intervals where the graph is positive/negative (above/below the x-axis)
10
Polynomial Graphs Homework 1. Fill in the missing information. Polynomial Function Name (degree) Name (terms) End Behavior 𝑓(𝑥) = 3𝑥2 − 5 𝑦 = −𝑥4 + 6𝑥 − 1 𝑔(𝑥) = 6𝑥 ℎ(𝑥) = 5𝑥2 − 2𝑥3 + 7𝑥 − 3 2. Identify the zeros of each function below. Be sure to state any multiplicity.
11
3. Use the given information to complete the missing columns. Table of Values Graph Key features of the function
The y-intercept is (0, 7). The zeros are located at x = 4 and x = 7. There is a relative minimum at (-5.5, 1.5) and at (5.5, -2.5). A relative maximum is located at (-1, 8.5). The polynomial is quartic.
4. Given the graph, state the intervals where the graph is
increasing/decreasing and where the graph is positive/negative.
X Y
X Y
12
Multiple Representations of Functions: Table & Graph 1. Write the equation for your assigned polynomial function below. 2. Use the given polynomial function to complete the table. Include values from various parts of
the function. (You may want to use your calculator to graph the function first to get an idea of what the graph looks like.)
3. Graph your function on the grid provided. Use a marker to help the function stand out.
Include the equation on the graph page. (Make sure that the important features of the function are graphed.)
4. Pair Share: Compare your table and graph with your partner. Discuss some of the similarities
and differences that you found between your two functions. Note any interesting similarities or differences between polynomial functions and functions that been discussed previously. Write down one thought.
x y
13
Equation: _________________________________
14
Multiple Representations of Functions: Reflection Part I: Number of Zeros: 1. Record at least two similarities and differences between the functions that have the same
number of zeros. Include examples.
Similarities Differences
2. What conclusions can be made about polynomial functions and the number of zeros they have?
Are there any special cases? If so, what conclusions can be made about the special cases? Part II: End Behavior: 3. Record at least two similarities and differences between the functions that have the same end
behavior. Include examples.
Similarities Differences
4. What conclusions can be made about polynomial functions and their end behavior? Are there
any special cases? If so, what conclusions can be made about the special cases?
15
Polynomial Graphs and Zeros The degree of a polynomial function gives a lot of information…
𝒚 = 𝒂𝒙 … 𝒚 = 𝒂𝒙𝟐 … 𝒚 = 𝒂𝒙𝟑 … 𝒚 = 𝒂𝒙𝟒 … 𝒚 = 𝒂𝒙𝟓 … Type of Polynomial Function
LINEAR
QUADRATIC
CUBIC
QUARTIC
QUINTIC
Domain Range
Maximum number of solutions/zeros (this is equal to the degree of the polynomial)
Maximum number of turns in the graph (this is one less than the degree of the polynomial)
Possible shape of the graph Positive a Negative a
End behavior Positive a Negative a
16
Polynomial Graph Matching Homework Part 1: Match each equation to its graph, zeros and end behavior
1) y = .8x2 + 3x - 8 2) y = 4.5x - 3 3) y = -2x5 + 4x4 + 3x3 - 5x2 + 6x + 20
4) y = (x+2)(x-3)(x+5)(x-8) 5) y = x3 - 8x2 + 10x + 10 6) y = -2x3 + 4x2 + 4x + 15
A. B. C.
D. E. F. # of zeros: End Behavior:
4 real roots 3 real roots
1 real root, 2 imaginary roots
(∞, -∞) (∞,∞) (-∞,∞)
1 real root, 4 imaginary roots
1 real root
2 real roots (∞, -∞) (-∞,∞) (∞,∞)
Write the equation Graph letter # of zeros End beavior
Created by: Cortella/Taylor, 2009
17
Zeros of a Polynomial Function Part 1: Look at the graph and state the x-intercepts; watch out for repeated roots!
x-intercepts: ________________________ x-intercepts: _________________________ equation: ____________________________ equation: ___________________________
x-intercepts: ________________________ equation: ____________________________ Part 2: Use the calculator to find any exact roots.
Zeros: _________________ Zeros: _________________
Factored form: ______________________ Factored form: ______________________
Zeros: _________________
Factored form: ______________________
18
Graph each function.
1.
y = -2 x2 - 9( ) x + 4( )
2. y = (x2 - 4)(x+3)
3. y = -1(x2-9)(x2-4)
4.
y =1
4x + 2( ) x -1( )
2
5.
y =1
5x - 3( )
2x +1( )
2
6.
y = x +1( )3x - 4( )
19
Math 3 Polynomial Parent Functions Linear Quadratic Cubic Quartic
Function Equation State the type of Function
Sketch the function Words: The graph moved… (compare to the parent function)
𝑦 = 𝑥3 − 3
__________________
𝑦 = (𝑥 + 5)2
__________________
𝑦 = (−3𝑥 + 1)4
__________________
𝑦 = −𝑥 + 6
__________________
20
Function Equation State the type of Function
Sketch the function Words: The graph moved… (compare to the parent function)
𝑦 = −(𝑥 + 4)3
__________________
𝑦 = −𝑥4 − 2
__________________
𝑦 = − (1
4𝑥 + 1)
2
+ 3
__________________
𝑦 = −2𝑥 + 5
__________________
𝑦 = −3𝑥2
__________________
21
Function Equation Parent Name
Graph the function Words: The graph moved… (compare to the parent function)
𝑦 = (−2𝑥 − 1)2 + 4
__________________
𝑦 = −(𝑥 + 1)3 + 2
__________________
General Form of a function 𝒇(𝒙) = 𝒂𝒇(𝒃𝒙 − 𝒉) + 𝒌
Summarize the different types of transformations When a > 1: _______________________________________________________________
When 0 < a < 1: _______________________________________________________________
When a is negative: _______________________________________________________________
When b > 1: _______________________________________________________________
When 0 < b < 1: _______________________________________________________________
When b is negative: _______________________________________________________________
When h is added: _______________________________________________________________
When h is subtracted: _______________________________________________________________
When k is added: _______________________________________________________________
When k is subtracted: _______________________________________________________________
22
Polynomial Transformations: Check for Understanding 1. Write an equation that will move the graph of the function y=x4 right 4 units and reflect over the x-axis.
2. The equation y = (x+3)2 – 2 moves the parent function y = x2 right 3 units and down 2 units. True or False
3. Write an equation that will move the graph of the function y = x3 down 7 units with a horizontal stretch of 3.
4. The equation y = (x-8)2 + 5 moves the parent function y = x2 right 8 units and down 5 units. True or False
5. Write an equation that will move the graph of the function y=x4 left 2 units and up 6 units with a reflection across the y-axis.
6. Which equation will shift the graph of y = x2 left 5 units and up 6 units?
a. y = (x+6)2 -5 b. y = (x+5)2 -6 c. y = (x+5)2 +6 d. y = (x-5)2 +6
7. Write an equation that will move the graph of the function y=x4 right 3 units up 2 units with a vertical stretch by 1/2.
8. Which equation will shift the graph of y = x2 right 8 units and down 4 units?
a. y = (x+8)2 -4 b. y = (x+4)2 -8 c. y = (x-4)2 +8 d. y = (x-8)2 -4
23
Transformations of Polynomial Functions Homework
1. Write the equation for the graph of function g(x), obtained by shifting the graph of f (x) = x² three units left, stretching the graph vertically by a factor of two, reflecting that result over the x-axis, and then translating the graph up four units.
2. Describe the transformations that would produce the graph of the second function from the graph of the first function.
a. 𝑓(𝑥) = 𝑥2 becomes 𝑓(𝑥) = (𝑥 − 3)2 + 5
b. 𝑓(𝑥) = 𝑥3 becomes 𝑓(𝑥) = −3𝑥3 − 1
c. 𝑓(𝑥) = 𝑥4 becomes 𝑓(𝑥) =1
2(𝑥 + 1)4 − 3
d. 𝑓(𝑥) = 𝑥2 becomes 𝑓(𝑥) = −2(3𝑥 − 2)2 + 5
3. Write the equation for the graph of function g(x), obtained by shifting the graph of f (x) = x4 two units right and up four units.
4 5
6 7
24
Polynomial Equations and Models
25
26
Polynomial Word Problems Practice and Homework 1. At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete
block with a volume of 945 cubic meters. The block’s dimensions are x meters high by 12x – 15 meters long by 12x – 21 meters wide. What is the height of the block?
2. You are designing a chocolate mold shaped like a hollow rectangular prism for a candy manufacturer. The mold must have a thickness of 1 cm in all dimensions. The mold’s outer dimensions should also be in the ratio 1:3:6. What should the outer dimensions of the mold be if it is to hold 112 cubic centimeters of chocolate?
3. A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about 89.58 cubic feet. The tank’s walls will be one half inch thick and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follows:
The width should be 2 feet less than the length The height should be 8 fee more than the length
What should the outer dimensions of the tank be? (HINT: Volume of steel = Volume outside – volume inside)
4. From 1985 to 2003, the total attendance A (in thousands) at NCAA women’s basketball games and the number T of NCAA women’s basketball teams can be modeled by 𝐴 = −1.95𝑥3 + 70.1𝑥2 −188𝑥 + 2150 and 𝑇 = 14.8𝑥 + 725 where x is the number of years since 1985. Compare and contrast the two functions. Find the attendance and number of teams for the year 1998.
5. Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to be 5 inches less than the length. What should the dimensions of the prism be if you want to use all of your clay?
6. The price p (in dollars) that a radio manufacturer is able to charge for a radio is given by 𝑝 = 40 −4𝑥2 where x is the number of radios produced in millions. It costs the company $15 to make a radio.
a) Write an expression for the company’s total revenue in terms of x b) Write a function for the company’s profit P by subtracting the total cost to make x radios from the
expression in part a c) Currently the company produces 1.5 million radios and makes a profit of $24,000,000. What lesser
number of radios can the company produce to make the same profit?
27
7. CHALLENGE: The profit P (in millions of dollars) for a DVD manufacturer can be modeled by 𝑃 =−6𝑥3 + 72𝑥 where x is the number of DVDs produced (in millions). Show that 2 million DVDs is the only production level for the company that yields a profit of $96,000,000.
8. A platform shaped like a rectangular prism has dimensions (x – 2) feet by (3 – 2x) feet by (3x + 4) feet. Explain why the volume of the platform cannot be 7/3 cubic feet.
BONUS
28
WRITING EQUATIONS OF POLYNOMIALS Write the equation from the graph: 1. 2. 3. 1. _________________________ 2. __________________________ 3. ___________________________ 4. 5. 6. 4. _________________________ 5. __________________________ 6. ___________________________ 7. 8. 9.
7. _________________________ 8. __________________________ 9. ___________________________
29
Polynomial Long Division
Ex 1: (3x3 – 5x2 + 10x – 3) ÷ 3x + 1 Ex 2: (2x3 – 9x2 + 15) ÷ (2x – 5)
and the answer is: and the answer is:Copyright © Elizabeth -2011
All Rights Reserved
The steps in the process of long division were: 1) Divide (first term into first term)
2) Multiply (use all terms) 3) Subtract 4) Bring Down 5) REPEAT
Dividing Polynomials - EXAMPLES Dividing by a monomial 1. (-30x3y + 12x2y2 – 18x2y) ÷ (-6x2y)
30
Divide using Long Division 2. (6x2 – x – 7) ÷ (3x + 1) 3. (4x2 – 2x + 6)(2x – 3)-1 4. (4x3 – 8x2 + 3x – 8) ÷ (2x – 1) 5. (2x3 – 3x2 – 18x – 8) ÷ (x – 4)
6.
(2x4 + 3x3 + 5x -1) ¸ (x2 - 2x + 2)
31
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x – a). It can be used in place of the standard long division algorithm.
Example: (2x4 – 3x3 - 5x2 + 3x + 8) ÷ (x-2)
All the variables and their exponents are removed, leaving only a list of the coefficients: 2, -3, -5, 3, 8. (Note: if a power of x is missing from the polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the polynomial.) These numbers form the dividend. We form the divisor for the synthetic division using only the constant term (2) of the linear factor (x-2). (Note: If the divisor were (x+2), we would use a divisor of -2.)
The numbers representing the divisor and the dividend are placed into a division-like configuration. First, bring down the "2" that indicates the leading coefficient:
Multiply by the number on the left, and carry the result into the next column:
Add down the column:
Multiply by the number on the left, and carry the result into the next column:
Add down the column:
Multiply by the number on the left, and carry the result into the next column:
Add down the column:
32
Multiply by the number on the left, and carry the result into the next column:
Add down the column for the remainder:
The completed division is:
All numbers except the last become the coefficients of the quotient polynomial. Since we started with a 4th degree polynomial and divided it by a linear term, the quotient is a 3rd degree polynomial. The last entry in the result list (2) is the remainder. The quotient and remainder can be combined into one expression:
2x3 + x2 – 3x - 3 + 2/(x-2) (Note that no division operations were performed to compute the answer to this division problem.) The steps in the process of synthetic division were: 1) Bring Down 2) Multiply 3) Add 4) REPEAT from 2)
Divide using Synthetic Division 7. (2x2 + 3x – 4) ÷ (x – 2) 8. (x4 – 3x3 + 5x – 6) ÷ (x + 2) 9. (2x3 + 4x – 6) ÷ (x + 3) 10. (x4 – 2x3 + 6x2 –8x + 10) ÷ (x + 2)
(HONORS) 11. (6x4 – x3 + 3x + 5) / (2x + 1)
33
Synthetic Division when the coefficient of x in the divisor ≠ 1 HONORS
Divide:
Step 1: Factor out the coefficient of x in the denominator.
(4𝑥3 − 8𝑥2 − 𝑥 + 5
2(𝑥 −12)
)
1
2(
4𝑥3 − 8𝑥2 − 𝑥 + 5
𝑥 −12
)
Step 2:, Set up the synthetic division, ignoring the ½ that
was factored out.
Step 3: Once the problem is set up correctly, bring the
leading coefficient (first number) straight down.
Step 4: Multiply the number in the division box with the
number you brought down and put the result in the next
column.
Step 5: Add the two numbers together and write the result
in the bottom of the row.
Step 6: Repeat steps 3 and 4 until you reach the end of the
problem.
Step 7: Multiply everything by the ½ which was factored
out in Step 1.
1
2(4𝑥2 − 6𝑥 − 4 +
3
𝑥−1
2
)
2𝑥2 − 3𝑥 − 2 +3
2𝑥 − 1
is the final answer
34
Long Division Practice Divide each of the polynomials using long division. 1. (4x2 – 9) ÷ (2x + 3) 2. (x2 – 4) ÷ (x + 4) 3. (2x2 + 5x – 3) ÷ (x + 3) 4. (2x2 + 5x – 3) ÷ (x – 3) 5. x4 – 1 6. x4 – 9 x2 – 1 x2 + 3 Synthetic Division Practice 7. (3x2 – 13x – 10) ÷ (x – 5) 8. (3x2 – 13x – 10) ÷ (x + 5) 9. (11x + 20x2 + 12x3 + 2) ÷ (3x + 2) 10. (12x3 + 2 + 11x + 20x2) ÷ (2x + 1)
35
Polynomial Division Worksheet
Divide using Long Division
1. (3y3 + 2y2 – 32y + 2) / (y – 3)
2. (2b3 + b2 – 2b + 3) / (b + 1)
3. (2c3 – 3c2 + 3c – 4) / (c – 2)
4. (3x3 – 2x2 + 2x – 1) / (x – 1)
Divide using Synthetic Division
5. (t4 – 2t3 + t2 – 3t + 2) / (t – 2)
6. (3r4 – 6r3 – 2r2 + r – 6) / (r + 1)
7. (z4 – 3z3 – z2 – 11z – 4) / (z – 4)
8. (2b3 – 11b2 + 12b + 9) / (b – 3)
9. (6s3 – 19s2 + s + 6) / (s – 3)
10. (x3 + 2x2 – 5x – 6) / (x – 2)
11. (x3 + 3x2 – 7x + 1) / (x – 1)
12. (n4 – 8n3 + 54n + 105) / (n – 5)
13. (2x4 – 5x3 + 2x – 3) / (x – 1)
14. (z5 – 6z3 + 4x2 – 3) / (z – 2)
Divide using long division:
15. (4s4 – 5s2 + 2s + 3) / (2s – 1) 16. (2x3 – 3x2 – 8x + 4) / (2x + 1)
Divide using Synthetic Division to factor completely
17. (2x3 + x2 – 22x + 24) / (2x – 3) 18. (6j3 – 19j2 + j + 6) / (3j – 2)
36
Remainder Theorem Divide p(x) = 𝑥3 − 7𝑥 − 6 by the linear term (x-4) Therefore:
𝑥3 − 7𝑥 − 6
𝑥 − 4 =
Multiply both sides by (x-4): Since p(x) = 𝑥3 − 7𝑥 − 6
What is the value of p(4)? What conclusion can you make about the remainder?
37
Remainder Theorem The value of the polynomial p(x) at x=a is the same as the remainder you get when you divide the polynomial by x-a.
To evaluate a polynomial p(x) at x=a, use synthetic division to divide the polynomial by x=a. The remainder is p(a)
Use the Remainder Theorem and Synthetic Division do find 𝑓(4) where 𝑓(𝑥) = −𝑥3 + 8𝑥2 + 63 The remainder theorem tells us that if we divide f(x) by (x-4) the remainder will be equal to f(4). Factor Theorem 𝑝(𝑎) = 0 if and only if x-a is a factor of 𝑝(𝑥).
If you divide a polynomial by x=a and get a zero remainder, then not only is x=a a zero of the polynomial, but x-a is a factor of the polynomial.
Determine whether x+4 is a factor of each polynomial A. 𝑓(𝑥) = 𝑥2 + 6𝑥 + 8 B. 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 6𝑥 − 7
38
Use the Remainder Theorem 1) Is (x – 1) a factor of x3 + 2x2 – 2x – 1? 2) Is (x + 2) a factor of 4x2 + 13x + 10? 3) What is the remainder when 3x3 + 10x2 + x – 6 is divided by x + 3 4) Is (x – 2) a factor of 4x2 + 13x + 10? 5) What is the remainder when 3x3 + 10x2 + x – 6 is divided by x – 1? Find the zeros using the given information 1) Find all the zeros of 𝑓(𝑥) = 𝑥3 − 4𝑥2 + 𝑥 + 6 given that x + 1 is a factor. 2) Solve for all the solutions of 2𝑥3 − 5𝑥2 + 𝑥 + 2 = 0 given that 2 is a solution.
3) Find all the zeros of 𝑔(𝑥) = 2𝑥3 + 3𝑥2 + 8𝑥 + 12 if −3
2 is a root.
39
Zeros & Remainder Theorem, Factor Theorem Worksheet
#1-6 Find the zeros of the polynomials and state the multiplicity of each zero: 1. f(x) = ( x + 4 )3 ( 3x – 4 )
2. f(x) = 2x5 – 8x4 – 10x3
3. f(x) = ( 9x2 – 25)4 ( x2 + 16)
4. f(x) = (x2 + x – 2)2 (x2 – 4)
5. f(x) = x (x + 2)3 (x – 5)
6. f(x) = 2x (x+3)2(4x – 1)
#7&8 Write a polynomial equation in standard from having the given roots:
7. 2, 3,- 1
8. 1 multiplicity 2, 0
9. -2 multiplicity 2, 1, 2
#10-15 Use synthetic division to show that c is a zero of f(x).
10. f(x) = 3x4 + 8x3 – 2x2 – 10x + 4 ; c = -2
11. f(x) = 4x3 – 9x2 – 8x – 3 ; c = 3
12. f(x) = 2x3 + 5x2 – 4x – 3 ; c = 1
13. f(x) = 2x4 + x3 – 14x2 + 5x + 6 ; c = -3
14. f(x)= 4x3 – 6x2 + 8x – 3; c = ½
15. f(x) = 27x4 – 9x3 + 3x2 + 6x + 1 ; c = -1/3
16. Factor f(x) = 9x3 + 6x2 – 3x if you know (x+1) is a factor.
17. Factor f(x) = x3 –2x2 – 9x + 18 if you know (x+3) is a factor.
40
18. Factor y = x3 – 4x2 – 3x +18 if you know that (x+2) is a factor.
19. Show that –3 is a zero of multiplicity 2 of the polynomial function
P(x)= x4 + 7x3 + 13x2 – 3x –18 and express P (x) as a product of linear factors.
20. Show that –1 is a zero of multiplicity 4 of the polynomial function
f(x)= x5 + x4 – 6x3 – 14x2 – 11x –3 and express f (x) as a product of linear factors.
21. Find a polynomial function of degree 4 such that both –2 and 3 are zeros of multiplicity 2.
22. Find a polynomial function of degree 5 such that –2 is a zero of multiplicity 3 and 4 is a zero of multiplicity 2.
23. Determine k so that that f(x ) = x3 + kx2– kx +10 is divisible by x +3.
24. Find k so that when x3 – x2 –kx + 10 is divided x –3 , the remainder is –2.
25. Find k so that when x3 – kx2– kx +1 is divided by x-2, the remainder =0
26. Determine k so that that f(x ) = 2kx3 + 2kx - 10 is divisible by x - 2.
27. SOLVE x3 + 4x2 – 5x = 0 completely.
41
Polynomial Modelling Word Problems When an equation is NOT given:
1. Define your variable(s) 2. If needed draw a picture. 3. Write an equation(s) to solve the problem. 4. State the solution. 5. Explain in words how you found the solution.
EXAMPLES: 1. The length of a rectangular pool is 4 yd longer than its width. The area of the pool is 60 yd. What are the dimensions of the pool? (6 x 10 yds) 2. A rectangle has a perimeter of 52 inches. Find the dimensions of the rectangle with maximum area. (13 x 13 in) 3. Find two consecutive negative integers whose product is 240. (-15 & -16) 4. Find two numbers who sum is 20 and whose product is a maximum (10 & 10).
42
Polynomial Word Problem Worksheet 1. Find two consecutive positive integers whose product is 462.
2. Find two numbers who difference is 8 and whose product is a minimum.
3. A rectangle has a perimeter of 48 inches. Find the dimensions of the rectangle with maximum area.
4. Find the negative integer whose square is 10 more than 3 times the integer.
5. One side of a rectangular garden is 2 yd less than the other side. If the area of the garden is 63 yd2, find the dimensions of the garden.
ANSWERS: 1) 21 & 22 2) 4 & -4 3) 12 x 12 in 4) -2 5) 7 x 9 yds.
43
6. Find two numbers who sum is -12 and whose product is a maximum.
7. Find 2 numbers whose sum is 36 and whose product is a maximum.
8. Find 2 numbers whose difference is 40 and whose product is a minimum.
9. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the maximum area.
10. Nick has 120 feet of fencing for a kennel. If his house is to be used as one side of the kennel, find the dimensions to maximize the area.
ANSWERS: 6) -6 & -6 7) 18 & 18 8) 20 & -20 9) 10 x 10 10) 60 x 30
44
Polynomial Functions Review 1. Complete the table below
Function Degree End Behavior
Domain and Range
A 𝑓(𝑥) = 3𝑥5 − 𝑥10
B 𝑔(𝑥) = −𝑥2 + 5𝑥 + 3
C ℎ(𝑥) = 3(𝑥 + 2)(𝑥 − 4)
D 𝑗(𝑥) = −2𝑥3 − 𝑥2 + 5𝑥 − 1
2. Evaluate the polynomial 𝑓(𝑥) = 3𝑥5 − 𝑥3 + 6𝑥2 − 𝑥 + 1 for 𝑥 = −2. Explain what your answer represents. 3. Find the zeros for the function 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 𝑥 − 3 4. Show whether -4 is a zero of 𝑔(𝑥) = 𝑥3 − 𝑥2 − 14𝑥 + 24 5. Use the graph to answer the following questions
a) Relative maximum:
b) Relative minimum:
c) Increasing interval:
d) Decreasing interval:
e) Domain:
f) Range:
g) End Behavior:
h) Zeros:
Find all the zeros 6. 𝑓(𝑥) = 2𝑥3 + 3𝑥2 − 39𝑥 − 20 and 4 is a zero 7. 𝑓(𝑥) = 𝑥4 + 3𝑥2 − 4 and 1 is a zero
45
Divide using long division 8. 𝑥3 − 3𝑥2 + 8𝑥 − 5 ÷ (𝑥 − 1) 9. 4𝑥3 − 12𝑥2 − 𝑥 + 15 ÷ (2𝑥 − 3) 10. Sketch a graph 𝑓(𝑥) = −4(𝑥 − 1)2(𝑥 − 3)(𝑥 + 8) 11. Write the function of 𝑥4 shifted 3 units down, 4 units left, a reflection over the x-axis and a horizontal compression by 3. 12. A cement walk of uniform width surrounds a rectangular swimming pool that is 10 m wide and 50 m long. Find the width of the walk if its area is 864 m2. 13. The number of eggs, f(x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by f(x) = 14x3 – 17x2 – 16x + 34. What is the abdominal width when there are 211 eggs? 14. A pyramid can be formed using equal-size balls. For example, 3 balls can
be arranged in a triangle, then a fourth ball placed in the middle on top of them. The function p(n) = )2)(1(
61 nnn gives the number of balls
in a pyramid, where n is the number of balls on each side of the bottom layer. (For the pyramid described above, n = 2. For the pyramid in the picture, n = 5.)
a. Evaluate p(2), p(3), and p(4). Sketch a picture of the pyramid that goes with each of these values. Check that your function values agree with your pyramid pictures.
b. If you had 1000 balls available and you wanted to make the largest possible pyramid using them, what would be the size of the bottom triangle, and how many balls would you use to make the pyramid? How many balls would be left over?
46
Sketch the graphs (no graphing calculator) 1. f(x) = (–1/5)(x + 3)(x + 5)(x – 2)2 2. f(x) = (–1/6)(x - 3)2(x - 1)(x + 2)2 Graphing Calculator Allowed
3. P(x) = (7x5 + 3x9 – 2x + 4) – (5x2 – 2x + 4)
a) Standard Form:_________________________________________________
b) Degree ___________________ c) Classify by the # of terms: __________________________ __________________4. Find p(3) for p(x) = -4x4 + 9x3 + 10x2 – 2x + 17 ________________________5. Is (x + 2) a factor of p(x) = x4 + 3x3 – 3x – 10 ? 6. Divide using long division (2x4 – 5x3 + 7x2 + 2x + 4) ÷ (2x – 3)
d.) Degree________________
e.) x- intercepts________________
f.) y-intercept _____________________
x
y
a.) Degree________________
b.) x- intercepts________________
c.) y-intercept _____________________
x
y
47
7. Write a polynomial function in standard form that has zeros at 2, -1, and 3 multiplicity 2?
8. Solve 0 = x3 – x2 – 11x + 3 given that -3 is a zero.
9. Is -3 a zero of p(x) = 2x4 + 9x3 – 7x + 10 ? Why or why not?
10. Is (x + 7) a factor of p(x) = x4 + 9x3 + 15x2 + 5x – 14 ? Why or why not?
11. Find p(3) for p(x) = 3x4 – 11x3 – x2 + 15x – 12 ?
12. Factor p(x) = 3x3 + 14x2 – 7x – 10 completely, given p(-5) = 0
13. Write the polynomial in factored form with zeros: 1 multiplicity 3, 0, -4 ?
14. Solve p(x) = x3 – 3x2 – 11x – 7 given that -1 is a zero.
15. Factor p(x) = 6x3 – 23x2 – 6x + 8 if (x – 4) is a factor.
48
16. Sketch the graph of p(x) = -1(x – 2)(x + 3)(x + 1) (no calc)
17. Solve p(x) = x3 – 11x2 + 36x – 36 if (x – 6) is a factor.
18. Solve p(x) = 15x3 – 119x2 – 10x + 16 if 8 is a zero.
19. Divide x4 − 3x3 + 18x2 −12x + 16 by x – 3 using long division.
20. One root of 2x3 −10x2 + 9x −4 = 0 is 4. Find the other roots.
21. If 3 + 2i is a zero of a polynomial, what has to be another zero?
22. Approximate to the nearest tenth the real zeros of f(x) = x3 −6x2 + 8x −2. (Use a calculator)
23. Write a polynomial function with zeros 1 and 2 (of multiplicity 3) in standard form.
24. Use synthetic division to find f(−2) if f(x) = 4x5 + 10x4 − 11x3 −22x2 + 20x + 10.
49
25. Factor: 2x3 + 15x2 −14x −48 if (x − 2) is a factor.
26. Determine if the degree of the functions below is even or odd. How many real zeros does each have?
a) b) c)
