algebra 2 5 polynomials and polynomial functions practice …rwright/algebra2/homework/chapter...

Algebra 2 5 Polynomials and Polynomial Functions Practice Problems

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5.1 Use Properties of Exponents Evaluate the expression. Tell which properties of exponents you used.

1. 33 ⋅ 32

2. 52

55

3. 34

3−2

Simplify the expression. Tell which properties of exponents you used. 4. (22𝑦3)5

5. (𝑤3𝑥−2)(𝑤6𝑥−1)

6. (3𝑎3𝑏5)−3

7. 3𝑐3𝑑

9𝑐𝑑−1

8. 2𝑎3𝑏−4

3𝑎5𝑏−2

9. 𝑥2𝑦−3

3𝑦2 ⋅𝑦2

𝑥−4

Describe and correct the error in simplifying the expression.

10. 𝑥10

𝑥2 = 𝑥5 11. (−3)2(−3)4 = 96

Write an expression that makes the statement true. 12. 𝑥15𝑦12𝑧8 = 𝑥4𝑦7𝑧11 ⋅ ?

Write an expression for the figure’s area or volume in terms of x. 13. 𝑉 = 𝜋𝑟2ℎ

Mixed Review

14. (4.10) Write a quadratic function in vertex form whose graph has vertex (-4, 1) and passes through (-2, 5).

15. (4.8) Solve 𝑥2 − 2𝑥 − 35 = 0 using any algebraic method.

16. (4.6) Solve 3𝑥2 − 7 = −31 using any algebraic method.

17. (3.6) Multiply [5 0

−4 1] [

−3 26 2

].

18. (3.2) Solve {4𝑥 −2𝑦 = −16

−3𝑥 +4𝑦 = 12

19. (2.4) Write an equation of the line that passes through the point (3, -1) and has the slope 𝑚 = −3.

20. (1.7) Solve |𝑓 − 5| = 3.

5.2 Evaluate and Graph Polynomial Functions 1. Identify the degree, type, leading coefficient, and constant term of the polynomial function 𝑓(𝑥) = 6 +

2𝑥2 − 5𝑥4.

2. Decide whether the function is a polynomial function. If so, write it is standard form and state its degree,

type, and leading coefficient. 𝑔(𝑥) = 𝜋𝑥4 + √6

Use direct substitution to evaluate the polynomial function for the given value of x. 3. 𝑓(𝑥) = 5𝑥3 − 2𝑥2 + 10𝑥 − 15; 𝑥 = −1 4. ℎ(𝑥) = 𝑥 +

1

2𝑥4 −

3

4𝑥3 + 10; 𝑥 = −4

Use synthetic substitution to evaluate the polynomial function for the given value of x. 5. 𝑔(𝑥) = 𝑥3 + 8𝑥2 − 7𝑥 + 35; 𝑥 = −6 6. ℎ(𝑥) = −7𝑥3 + 11𝑥2 + 4𝑥; 𝑥 = 3

Describe the degree and leading coefficient of the polynomial function whose graph is shown.

𝑥

𝑥

2


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7. Describe the end behavior of the graph of the polynomial function by completing these statements: 𝒇(𝒙) →

_________ as x → −∞ and 𝒇(𝒙) → _______ as x → +∞.

8. 𝑓(𝑥) = −𝑥6 + 4𝑥3 − 3𝑥 9. 𝑓(𝑥) = −6𝑥5 + 14𝑥2 + 20

10. Write a polynomial function f of degree 5 such that the end behavior of the graph of x is given by f(x) → +∞

as x → −∞ and f(x) → −∞ as x → +∞. Then graph the function to verify your answer.

Graph the polynomial function. 11. 𝑓(𝑥) = 𝑥4 − 2

12. 𝑓(𝑥) = 𝑥5 + 𝑥

13. 𝑓(𝑥) = −𝑥4 + 3𝑥3 − 𝑥 + 1

Word problems 14. From 1992 to 2003, the number of people in the United States who participated in skateboarding can be

modeled by

𝑆 = −0.0076𝑡4 + 0.14𝑡3 − 0.62𝑡2 + 0.52𝑡 + 5.5

where S is the number of participants (in millions) and t is the number of years since 1992. Graph the

model. Then use the graph to estimate the first year that the number of skateboarding participants was

greater than 8 million.

15. The weights of Sarus crane chicks S and hooded crane chicks H (both in grams) during the 10 days

following hatching can be modeled by

𝑆 = −0.122𝑡3 + 3.49𝑡2 − 14.6𝑡 + 136

𝐻 = −0.115𝑡3 + 3.71𝑡2 − 20.6𝑡 + 124

where t is the number of days after hatching.

a. According to the models, what is the difference in eight between 5-day-old Sarus crane chicks and

hooded crane chicks?

b. Sketch the graphs of the two models on the same coordinate plane.

c. A biologist finds that the weight of a crane chick 3 days after hatching is 130 grams. What species of

crane is the chick more likely to be? Explain how you found your answer.

Mixed Review

16. (5.1) Simplify the expression 𝑤−2

𝑤6

17. (5.1) Evaluate the expression (4−2)3

18. (4.2) Write the quadratic equation in standard form 𝑦 = (𝑥 + 4)(𝑥 + 3)

19. (4.2) Write the quadratic equation in standard form 𝑦 = −(𝑥 + 6)2 + 10

20. (3.8) Find the inverse of the matrix [−2 3−3 4

]


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5.3 Add, Subtract, and Multiply Polynomials 1. When you add or subtract polynomials, you add or subtract the coefficients of ___________________?

Find the sum or difference. 2. (4𝑦2 + 9𝑦 − 5) − (4𝑦2 − 5𝑦 + 3)

3. (5𝑐2 + 7𝑐 + 1) + (2𝑐3 − 6𝑐 + 8)

4. (𝑥4 − 𝑥3 + 𝑥2 − 𝑥 + 1) + (𝑥 + 𝑥4 − 1 − 𝑥2)

Find the product of the polynomials. 5. 5𝑥2(6𝑥 + 2)

6. (2𝑎 − 3)(𝑎2 − 10𝑎 − 2)

7. (3𝑦2 + 6𝑦 − 1)(4𝑦2 − 11𝑦 − 5)

8. (𝑥 + 1)(𝑥 − 7)(𝑥 + 3)

9. (𝑏 − 2)(2𝑏 − 1)(−𝑏 + 1)

10. (3𝑞 − 8)(−9𝑞 + 2)(𝑞 − 2)

11. (2𝑐 + 5)2

12. (2𝑎 + 9𝑏)(2𝑎 − 9𝑏)

Verify the special by multiplying. 13. (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2

Word problems 14. Look at the following polynomial factorizations.

𝑥2 − 1 = (𝑥 − 1)(𝑥 + 1)

𝑥3 − 1 = (𝑥 − 1)(𝑥2 + 𝑥 + 1)

𝑥4 − 1 = (𝑥 − 1)(𝑥3 + 𝑥2 + 𝑥 + 1)

a. Factor 𝑥5 − 1 and 𝑥6 − 1 into the product of 𝑥 − 1 and another polynomial. Check your answers by

multiplying.

b. In general, how can 𝑥𝑛 − 1 be factored? Show that this factorization works by multiplying the

factors.

15. The equation 𝑃 = 0.00267𝑠𝐹 gives the power P (in horsepower) needed to keep a certain bicycle moving at

speed s (in miles per hour), where F is the force (in pounds) of road and air resistance. On level ground, the

equation

𝐹 = 0.0116𝑠2 + 0.789

models the force F. Write a model (in terms of s only) for the power needed to keep the bicycle moving at

speed s on level ground. How much power is needed to keep the bicycle moving at 10 miles per hour?

Mixed Review 16. (5.2) Use synthetic substitution to find 𝑓(2)

when 𝑓(𝑥) = 𝑥5 + 3𝑥2 − 6𝑥 − 30

17. (5.1) Simplify the expression (𝑝3𝑞2)−1

18. (4.4) Factor 3𝑛2 + 7𝑛 + 4

19. (4.4) Factor 9𝑥2 − 16

20. (4.3) Factor 𝑥2 − 7𝑥 + 10

5.4 Factor and Solve Polynomial Equations 1. The expression 8𝑥6 + 10𝑥3 − 3 in _________ form because it can be written as 2𝑢2 + 5𝑢 − 3 where 𝑢 = 2𝑥3.

Factor the polynomial completely. 2. 𝑐3 + 9𝑐2 + 18𝑐

3. 𝑦3 − 64

4. 8𝑐3 + 343

5. −5𝑧3 + 320

6. 𝑦3 − 7𝑦2 + 4𝑦 − 28

7. 3𝑚3 − 𝑚2 + 9𝑚 − 3

8. 4𝑐3 + 8𝑐2 − 9𝑐 − 18

9. 𝑎4 + 7𝑎2 + 6

10. 32𝑧5 − 2𝑧

11. 15𝑥5 − 72𝑥3 − 108𝑥

12. 36𝑎3 − 15𝑎2 + 84𝑎 − 35

13. 8𝑦6 − 38𝑦4 − 10𝑦2

Find all the solutions of the equation. 14. 18𝑠3 = 50𝑠 15. 𝑚3 + 6𝑚2 − 4𝑚 − 24 = 0


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Word problems 16. Suppose you have 250 cubic inches of glass 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?

17. Use the diagram to complete parts (a) – (c).

a. Explain why 𝑎3 − 𝑏3 is equal to the sum of the volumes of solid I,

solid II, and solid III.

b. Write an algebraic expression for the volume of each of the three

solids. Leave your expression in factored form.

c. Use the results from parts (a) and (b) to derive the factoring

pattern for 𝑎3 − 𝑏3.

Mixed Review 18. (5.3) Simplify (𝑥2 − 3𝑥 + 5) − (−4𝑥2 + 8𝑥 + 9)

19. (5.3) Simplify (𝑤 + 4)(𝑤2 + 6𝑤 − 11)

20. (5.2) Use synthetic substitution to evaluate 𝑓(𝑥) = 8𝑥4 + 12𝑥3 + 6𝑥2 − 5𝑥 + 9 for 𝑥 = −2.

5.5 Apply the Remainder and Factor Theorems Divide using polynomial long division.

1. (𝑥2 + 𝑥 − 17) ÷ (𝑥 − 4)

2. (𝑥3 + 3𝑥2 + 3𝑥 + 2) ÷ (𝑥 − 1)

3. (3𝑥3 + 11𝑥2 + 4𝑥 + 1) ÷ (𝑥2 + 𝑥)

4. (5𝑥4 − 2𝑥3 − 7𝑥2 − 39) ÷ (𝑥2 + 2𝑥 − 4)

Divide using synthetic division. 5. (2𝑥2 − 7𝑥 + 10) ÷ (𝑥 − 5)

6. (𝑥2 + 8𝑥 + 1) ÷ (𝑥 + 4)

7. (𝑥3 − 5𝑥2 − 2) ÷ (𝑥 − 4)

8. (𝑥4 − 5𝑥3 − 8𝑥2 + 13𝑥 − 12) ÷ (𝑥 − 6)

Given polynomial 𝒇(𝒙) and a factor of 𝒇(𝒙), factor 𝒇(𝒙) completely. 9. 𝑓(𝑥) = 𝑥3 − 10𝑥2 + 19𝑥 + 30; 𝑥 − 6

10. 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 40𝑥 − 64; 𝑥 − 8

11. 𝑓(𝑥) = 𝑥3 + 2𝑥2 − 51𝑥 + 108; 𝑥 + 9

Given the polynomial f and a zero of f, find the other zeros. 12. 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 21𝑥 − 18; −3

13. 𝑓(𝑥) = 10𝑥3 − 81𝑥2 + 71𝑥 + 42; 7

14. 𝑓(𝑥) = 2𝑥3 − 10𝑥2 − 71𝑥 − 9; 9

Word problems 15. The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled by 𝑃 = −𝑥3 + 4𝑥2 + 𝑥 where

x is the number of T-shirts produced (in millions). Currently, the company produces 4 million T-shirts and

makes a profit of $4,000,000. What lesser number of T-shirts could the company produce and still make the

same profit?

Mixed Review 16. (5.4) Find the solutions of 𝑦3 − 5𝑦2 = 0.

17. (5.4) Factor completely 16𝑥3 − 44𝑥2 − 42𝑥

18. (5.3) Simplify (𝑤 − 9)2

19. (5.2) Describe the end behavior of the graph of 𝑓(𝑥) = 10𝑥4 by completing this statement: 𝑓(𝑥) → _______ as

x → −∞ and 𝑓(𝑥) → ________ as x → +∞.

20. (4.8) Use the quadratic formula to solve 𝑥2 − 6𝑥 + 7 = 0.


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5.6 Find Rational Zeros List the possible rational zeros of the function using the rational zero theorem.

1. 𝑓(𝑥) = 𝑥3 − 3𝑥 + 28

2. 𝑓(𝑥) = 2𝑥4 + 6𝑥3 − 7𝑥 + 9

3. 𝑔(𝑥) = 4𝑥5 + 3𝑥3 − 2𝑥 − 14

4. ℎ(𝑥) = 8𝑥4 + 4𝑥3 − 10𝑥 + 15

Find all the real zeros of the function. 5. 𝑓(𝑥) = 𝑥3 − 12𝑥2 + 35𝑥 − 24

6. 𝑔(𝑥) = 𝑥3 − 31𝑥 − 30

7. ℎ(𝑥) = 𝑥4 + 7𝑥3 + 26𝑥2 + 44𝑥 + 24

8. 𝑓(𝑥) = 𝑥4 + 2𝑥3 − 9𝑥2 − 2𝑥 + 8

Use the graph to shorten the list of possible rational zeros of the function. Then find all the real zeros of the function.

9. 𝑓(𝑥) = 4𝑥3 − 20𝑥 + 16

10. 𝑓(𝑥) = 6𝑥3 + 25𝑥2 + 16𝑥 − 15

Find all the real zeros of the function.

11. 𝑓(𝑥) = 3𝑥3 + 4𝑥2 − 35𝑥 − 12

12. 𝑔(𝑥) = 2𝑥3 + 5𝑥2 − 11𝑥 − 14

13. ℎ(𝑥) = 2𝑥4 − 𝑥3 − 7𝑥2 + 4𝑥 − 4

14. ℎ(𝑥) = 2𝑥5 + 5𝑥4 − 3𝑥3 − 2𝑥2 − 5𝑥 + 3

Word problem 15. At a factory, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism

with a height 4 inches greater than the length of each side of its square base. Each mold holds 63 cubic

inches of molten glass. What are the dimensions of the mold?

Mixed Review 16. (5.5) Divide using synthetic division (4𝑥2 − 13𝑥 − 5) ÷ (𝑥 − 2)

17. (5.5) Divide using polynomial long division (3𝑥2 − 11𝑥 − 26) ÷ (𝑥 − 5)

18. (5.4) Find the real-number solutions of the equation by factoring 𝑔3 + 3𝑔2 − 𝑔 − 3

19. (5.3) Simplify (𝑥 + 5)(𝑥 − 5)

20. (3.2) Solve the system of equations {3𝑥 + 𝑦 = 162𝑥 − 3𝑦 = −4

5.7 Apply the Fundamental Theorem of Algebra 1. Copy and complete: For the equation (𝑥 − 1)2(𝑥 + 2) = 0, a(n) _______ solution is 1 because the factor 𝑥 − 1

appears twice.

Identify the number of solutions or zeros. 2. 9𝑡6 − 14𝑡3 + 4𝑡 − 1 = 0 3. 𝑓(𝑥) = 16𝑥 − 22𝑥3 + 6𝑥6 + 19𝑥5 − 3

Find all the zeros of the polynomial function. 4. ℎ(𝑥) = 𝑥3 + 5𝑥2 − 4𝑥 − 20

5. 𝑓(𝑥) = 𝑥4 + 𝑥3 + 2𝑥2 + 4𝑥 − 8

6. 𝑔(𝑥) = 𝑥4 − 2𝑥3 − 3𝑥2 + 2𝑥 + 2

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

7. −2, 1, 3

8. 3i, 2 – i

9. −4, 1, 2 − √6


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Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.

10. ℎ(𝑥) = 𝑥5 − 2𝑥3 − 𝑥2 + 6𝑥 + 5 11. 𝑓(𝑥) = 𝑥7 + 4𝑥4 − 10𝑥 + 25

Use a graphing calculator to graph the function. Then use the zero feature to approximate the real zeros of the function.

12. ℎ(𝑥) = 𝑥4 − 5𝑥 − 3 13. 𝑔(𝑥) = 𝑥5 − 16𝑥3 − 3𝑥2 + 42𝑥 + 30

Determine the numbers of positive real zeros, negative real zeros, and imaginary zeros for the function with the given degree and graph. Explain your reasoning.

14. Degree: 3

Word problem

15. From 1990 to 2003, the number N of inland lakes in Michigan infested with zebra mussels can be modeled

by the function

𝑁 = −0.028𝑡4 + 0.59𝑡3 − 2.5𝑡2 + 8.3𝑡 − 2.5

where t is the number of years since 1990. In which year did the number of infested inland lakes first reach

120?

Mixed Review 16. (5.6) List the possible rational zeros of the function. 𝑔(𝑥) = 𝑥3 − 4𝑥2 + 𝑥 − 10

17. (5.6) Find all the real zeros of 𝑔(𝑥) = 2𝑥3 − 7𝑥2 + 9

18. (5.5) Divide using polynomial long division (8𝑥2 + 34𝑥 − 1) ÷ (4𝑥 − 1)

19. (5.4) Factor 𝑥3 + 𝑥2 + 𝑥 + 1

20. (5.3) Simplify (3𝑥2 − 5) + (7𝑥2 − 3)

5.8 Analyze Graphs of Polynomial Functions Graph the function.

1. 𝑓(𝑥) = (𝑥 − 2)2(𝑥 + 1)

2. 𝑔(𝑥) =1

3(𝑥 − 5)(𝑥 + 2)(𝑥 − 3)

3. ℎ(𝑥) = 4(𝑥 + 1)(𝑥 + 2)(𝑥 − 1)

4. 𝑓(𝑥) = 2(𝑥 + 2)2(𝑥 + 4)2

5. 𝑔(𝑥) = (𝑥 − 3)(𝑥2 + 𝑥 + 1)

Describe and correct the error in graphing f. 6. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 1)2

Estimate the coordinates of each turning point an dstate whether each corresponds to a local maximum or a local minimum. Then estimate all real zeros and determind the least degree the function can have.


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7.

8.

9.

10.

11. Which point is a local maximum of the function 𝑓(𝑥) = 0.25(𝑥 + 2)(𝑥 − 1)2?

A. (-2, 0) B. (-1, 1) C. (1, 0) D. (2, 1)

12. Why is the adjective local, used to describe the maximums and minimums of cubic functions, not required

for quadratic functions?

Graph the function. Then identify its domain and range. 13. 𝑓(𝑥) = 𝑥2(𝑥 − 2)(𝑥 − 4)(𝑥 − 5) 14. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 1)(𝑥 − 1)2(𝑥 − 2)2

Word problem 15. For a swimmer doing the breaststroke, the function

𝑆 = −241𝑡7 + 1060𝑡6 − 1870𝑡5 + 1650𝑡4 − 737𝑡3 + 144𝑡2 − 2.43𝑡

models the swimmer’s speed S (in meters per second) during one complete stroke, where t is the number

of seconds since the start of the stroke. Graph the function. According to the model, at what time during the

stroke is the swimmer going the fastest?

16. (5.7) Find all zeros of 𝑓(𝑥) = 𝑥4 − 6𝑥3 + 7𝑥2 + 6𝑥 − 8

17. (5.7) Find all zeros of 𝑓(𝑥) = 𝑥4 + 15𝑥2 − 16

18. (5.6) List all the possible rational zeros of ℎ(𝑥) = 2𝑥3 + 𝑥2 − 𝑥 − 18

19. (5.5) Completely factor 𝑓(𝑥) = 𝑥3 + 6𝑥2 + 5𝑥 − 12 given that 𝑥 + 4 is a factor.

20. (5.1) Simplify 𝑥−1𝑦2

𝑥2𝑦−1


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5.9 Write Polynomial Functions and Models 1. Copy and complete: When the x-values in a data set are equally spaced, the differences of consecutive y-

values are called ____________.

Write the cubic function whose graph is shown.

2. 3. Write a cubic function whose graph passes through the points.

4. (−2, 0), (−1, 0), (0, −8), (2, 0) 5. (−5, 0), (0, 0), (1, −12), (6, 0)

6. A student tried to write a cubic function whose graph has x-intercepts −1, 2,

5, and passes through (1, 3). Describe and correct the error in the student’s

calculation of the leading coefficient a.

Show that the nth-order differences for the given function of degree n are nonzero and constant.

7. 𝑓(𝑥) = −2𝑥2 + 5𝑥

8. 𝑓(𝑥) = 4𝑥2 − 9𝑥 + 2

9. 𝑓(𝑥) = 2𝑥5 − 3𝑥2 + 𝑥

Use finite differences and a system of equations to find a polynomial function that fits the data in the table.

10. 11. Word problems

12. How many points do you need to determine a quartic function? a quintic (fifth-degree) function? Justify

your answers.

13. Find a polynomial function that gives the number of diagonals d of a polygon with n sides.

14. The maximum number of regions R into which space can be divided by n intersecting spheres is given by

𝑅(𝑛) =1

3𝑛3 − 𝑛2 +

8

3𝑛. Show that this function has constant third-order differences.

Mixed Review 15. (5.8) Graph 𝑓(𝑥) = (𝑥 + 1)2(𝑥 − 1)(𝑥 − 3)

16. (5.8) Estimate the coordinates of each turning point and state whether each

corresponds to a local maximum or a local minimum. Then estimate all real

zeros and determine the least degree the function can have.

17. (5.7) Write a polynomial of least degree with the zeros 2, −i, and i

18. (5.6) Find all the real zeros of 𝑓(𝑥) = 𝑥3 − 5𝑥2 − 22𝑥 + 56

19. (5.4) Factor 𝑥3 + 8

20. (5.2) Describe the end behavior of the graph of the polynomial function by completing these statements:

𝑓(𝑥) → _______ as x → −∞ and 𝑓(𝑥) → _______ as x → +∞. 𝑓(𝑥) = −2𝑥3 + 7𝑥 − 4


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Chapter 5 Review Simplify.

1. (10𝑥8𝑦)(25𝑥3𝑦11)

2. (2𝑥10𝑦−5)3

3. 𝑟10 ⋅ 𝑡3 ÷ 𝑟3

Perform in the indicated operation. 4. (10𝑥3 + 2𝑥2 − 13𝑥 + 7) + (2𝑥3 + 25𝑥 − 17)

5. (𝑥2 + 4)(𝑥2 + 5𝑥 − 10)

6. (10𝑥3 − 3𝑥2 + 2𝑥 − 4) ÷ (𝑥2 + 5𝑥 − 1)

7. (10𝑥3 − 3𝑥2 + 2𝑥 − 4) ÷ (𝑥 + 4)

Factor the polynomial completely. 8. 2𝑥3 − 𝑥2 − 4𝑥 + 2

9. 3𝑥4 − 11𝑥2 − 4

10. 3𝑥3 + 24

11. Find the other zeros of 𝑓(𝑥) = 𝑥3 + 2𝑥2 − 5𝑥 − 6 given that one zero is 2. (Show work)

12. List the possible rational zeros of 𝑓(𝑥) = 4𝑥4 + 3𝑥3 − 5𝑥2 + 6𝑥 − 9.

13. Find all the zeros of 𝑓(𝑥) = 𝑥3 − 13𝑥 + 12. (Show work of verifying your answers.)

14. Find all the zeros of 𝑓(𝑥) = 𝑥3 − 𝑥2 + 2. (Show work of verifying your answers.)

15. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 (a = 1),

and zeros 1 − 𝑖 and 3.

Graph the function. 16. 𝑓(𝑥) = −(𝑥 − 1)(𝑥 + 1)(𝑥 − 2)

17. 𝑓(𝑥) = (𝑥 + 2)2(𝑥 − 2)

18. 𝑓(𝑥) = 𝑥4 − 𝑥 − 2

19. Using the following graph, identify the x-intercepts, local maximums, and local minimuims.

20. Show that the third-order differences for 𝑓(𝑥) = 𝑥3 + 𝑥2 − 𝑥 + 1 are nonzero and constant.

21. A cylinder has a height that is 3 units more than the radius. If the volume is 200π units3, find the length of

the radius.


of 12

Answers

5.1 1. 243; Product of Powers Property

2. 1

125; Quotient of Powers Property

3. 729; Quotient of Powers Property 4. 1024𝑦15; Power of a Power Property

5. 𝑤9

𝑥3; Product of Power Property;

Negative Exponent Property

6. 1

27𝑎9𝑏15; Negative Exponent Property;

Power of a Product Property

7. 𝑐2𝑑2

3; Quotient of Powers Property

8. 2

3𝑎2𝑏2; Quotient of Powers Property

9. 𝑥6

3𝑦3; Product of Powers Property;

Quotient of Powers Property 10. The exponents should be subtracted; 𝑥8 11. The base should not change; (−3)6 12. 𝑥11𝑦5𝑧−3

13. 𝜋𝑥3

2

14. 𝑦 = (𝑥 + 4)2 + 1 15. −5, 7

16. ±2√2𝑖

17. [−15 1018 −6

]

18. (-4, 0) 19. 𝑦 = −3𝑥 + 8 20. 2, 8

5.2 1. Degree: 4; Type: Quartic; Leading Coefficient: -5; Constant term: 6 2. Polynomial; Degree: 4; Type: Quartic; Leading coefficient: π 3. -32 4. 182 5. 149 6. -78 7. Degree: even; Leading coefficient: positive

8. −∞, −∞ 9. +∞, −∞

10. Sample: 𝑓(𝑥) = −𝑥5 − 2𝑥4 + 1

11.

12.

13.

14. ; 1998

15. 35.625 g;

; Sarus crane

16. 1

𝑤8 17.

1

4096

18. 𝑦 = 𝑥2 + 7𝑥 + 12 19. 𝑦 = −𝑥2 − 12𝑥 − 26

20. [4 −33 −2

]

5.3 1. like terms 2. 14𝑦 − 8 3. 2𝑐3 + 5𝑐2 + 𝑐 + 9 4. 2𝑥4 − 𝑥3 5. 30𝑥3 + 10𝑥2 6. 2𝑎3 − 23𝑎2 + 26𝑎 + 6 7. 12𝑦4 − 9𝑦3 − 85𝑦2 − 19𝑦 + 5 8. 𝑥3 − 3𝑥2 − 25𝑥 − 21 9. −2𝑏3 + 7𝑏2 − 7𝑏 + 2 10. −27𝑞3 + 132𝑞2 − 172𝑞 + 32

11. 4𝑐2 + 20𝑐 + 25 12. 4𝑎2 − 81𝑏2 13. Show work 14. (𝑥 − 1)(𝑥4 + 𝑥3 + 𝑥2 + 𝑥 + 1); (𝑥 − 1)(𝑥5 + 𝑥4 + 𝑥3 + 𝑥2 + 𝑥 + 1); (𝑥 − 1)(𝑥𝑛−1 + 𝑥𝑛−2 + ⋯ + 𝑥2 + 𝑥 + 1) 15. 𝑃 = 0.000031𝑠3 + 0.002107𝑠; about 0.05 horsepower 16. 2

17. 1

𝑝3𝑞2

18. (3𝑛 + 4)(𝑛 + 1) 19. (3𝑥 − 4)(3𝑥 + 4) 20. (𝑥 − 5)(𝑥 − 2)

5.4 1. quadratic 2. 𝑐(𝑐 + 3)(𝑐 + 6) 3. (𝑦 − 4)(𝑦2 + 4𝑦 + 16) 4. (2𝑐 + 7)(4𝑐2 + 14𝑐 + 49) 5. −5(𝑧 − 4)(𝑧2 + 4𝑧 + 16) 6. (𝑦 − 7)(𝑦2 + 4) 7. (3𝑚 − 1)(𝑚2 + 3) 8. (𝑐 + 2)(2𝑐 − 3)(2𝑐 + 3) 9. (𝑎2 + 1)(𝑎2 + 6)

10. 2𝑧(2𝑧 − 1)(2𝑧 + 1)(4𝑧2 + 1) 11. 3𝑥(𝑥2 − 6)(5𝑥2 + 6) 12. (12𝑎 − 5)(3𝑎2 + 7) 13. 2𝑦2(𝑦2 − 5)(4𝑦2 + 1)

14. 0, −12

3, 1

2

3 15. 2, −2, −6

16. 10 in. × 5 in. × 5 in.

17. The whole thing is 𝑎3 and the missing piece is 𝑏3; I: (𝑎)(𝑎)(𝑎 − 𝑏) II: (𝑎)(𝑏)(𝑎 − 𝑏) III: (𝑏)(𝑏)(𝑎 − 𝑏); 𝑎3 −𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2) 18. 5𝑥2 − 11𝑥 − 4 19. 𝑤3 + 10𝑤2 + 13𝑤 − 44 20. 75


of 12

5.5 1. 𝑥 + 5 +

3

𝑥−4

2. 𝑥2 + 4𝑥 + 7 +9

𝑥−1

3. 3𝑥 + 8 +−4𝑥+1

𝑥2+𝑥

4. 5𝑥2 − 12𝑥 + 37 +−122𝑥+109

𝑥2+2𝑥−4

5. 2𝑥 + 3 +25

𝑥−5

6. 𝑥 + 4 +−15

𝑥+4

7. 𝑥2 − 𝑥 − 4 +−18

𝑥−4

8. 𝑥3 + 𝑥2 − 2𝑥 + 1 +−6

𝑥−6

9. (𝑥 − 6)(𝑥 − 5)(𝑥 + 1) 10. (𝑥 − 8)(𝑥 + 2)(𝑥 + 4) 11. (𝑥 − 4)(𝑥 − 3)(𝑥 + 9)

12. −1, 6 13. −2

5,

3

2 14.

−4±√14

2

15. 1 million T-shirts 16. 0, 5 17. 2𝑥(2𝑥 − 7)(4𝑥 + 3)

18. 𝑤2 − 18𝑤 + 81 19. +∞, +∞

20. 3 ± √2

5.6 1. ±1, ±2, ±4, ±7, ±14, ±28

2. ±1, ±3, ±9, ±1

2, ±

3

2, ±

9

2

3. ±1, ±2, ±7, ±14, ±1

2, ±

7

2, ±

1

4, ±

7

4

4. ±1, ±3, ±5, ±15, ±1

2, ±

3

2, ±

5

2, ±

15

2,

±1

4, ±

3

4, ±

5

4, ±

15

4, ±

1

8, ±

3

8, ±

5

8, ±

15

8

5. 1, 3, 8 6. −5, −1, 6 7. −2, −1

8. −4, −1, 1, 2 9. 1,−1±√17

2

10. −3, −5

3,

1

2 11. −4, −

1

3, 3

12. −7

2, −1, 2 13. −2, 2

14. −3,1

2, 1

15. 3 in. × 3 in. × 7 in.

16. 4𝑥 − 5 +−15

𝑥−2

17. 3𝑥 + 4 +−6

𝑥−5

18. −3, −1, 1 19. 𝑥2 − 25 20. (4, 4)

5.7 1. repeated 2. 6 3. 6 4. −5, −2, 2 5. −2, 1, 2𝑖, −2𝑖

6. −1, 1, 1 + √3, 1 − √3 7. 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 5𝑥 + 6 8. 𝑓(𝑥) = 𝑥4 − 4𝑥3 + 14𝑥2 − 36𝑥 + 45 9. 𝑓(𝑥) = 𝑥4 − 𝑥3 − 18𝑥2 + 10𝑥 + 8

10. Positive: 2 or 0; Negative: 3 or 1; Imaginary: 4, 2, or 0 11. Positive: 2 or 0; Negative: 1; Imaginary: 6 or 4 12. 𝑥 ≈ −0.58, 𝑥 ≈ 1.9 13. 𝑥 ≈ −3.5, 𝑥 ≈ −1.1, 𝑥 = −1, 𝑥 ≈2.1, 𝑥 ≈ 3.6

14. Positive: 1, Negative: 2, Imaginary: 0 15. 1999 16. ±1, ±2, ±5, ±10

17. −1,3

2, 3 18. 2𝑥 + 9 +

8

4𝑥−1

19. (𝑥 + 1)(𝑥2 + 1) 20. 10𝑥2 − 8

5.8

1.

2.

3.

4.

5. 6. The intercepts should be at −2 and 1.

7. Max: (-0.3, 0.3), Min: (0.9, -1.3), Zeros: -0.75, 0, 1.4, Degree: 3 8. local maximum: (20.5, 22.4), local minimum: (1.5, 25.7); zero: 2.7, least degree: 3

9. local maximums: (1, 0), (3, 0), local minimum: (2, 22); zeros: 1, 3, least degree: 4 10. local maximums: (21.1, 0.8), (1.9, 8), local minimums: (22.2, 238), (0.3, 241), (2.8, 213); zeros: 22.6, 21.2, 1.5, 2.2, 3, least degree: 6 11. B 12. Cubic has no absolute max or min

13. , Domain: All real numbers, Range: All real numbers

14. ;

domain: all real numbers, range: y ≥

−21.3


of 12

15. ; 0.95 s

16. -1, 1, 2, 4 17. -1, 1, -4i, 4i

18. ±1, ±2, ±3, ±6, ±9, ±18, ±1

2, ±

3

2, ±

9

2

19. (𝑥 − 1)(𝑥 + 3)(𝑥 + 4)

20. 𝑦3

𝑥3

5.9 1. finite differences 2. 𝑦 = 0.5𝑥3 − 2𝑥2 + 0.5𝑥 + 3

3. 𝑦 =1

6𝑥3 −

1

3𝑥2 −

11

6𝑥 + 2

4. 𝑦 = 2𝑥3 + 2𝑥2 − 8𝑥 − 8

5. 𝑦 =2

5𝑥3 −

2

5𝑥2 − 12𝑥

6. x should be 1 and y should be 3; 𝑎 =3

8

7.

8.

9.

10. 𝑓(𝑥) = −4𝑥2 + 15𝑥 11. 𝑓(𝑥) = −0.5𝑥3 + 5𝑥2 − 2.5𝑥 + 3 12. 5; 6; there must be one more data point than the degree of the equation. 13. 𝑑 = 0.5𝑛2 − 1.5𝑛

14.

15. 16. local maximum: (−0.25, −2), local minimums: (−1.5, −5), (0.5, −2.4); zeros: −2, 1, least degree: 4 17. 𝑓(𝑥) = 𝑥3 − 2𝑥2 + 𝑥 − 2 18. −4, 2, 7 19. (𝑥 + 2)(𝑥2 − 2𝑥 + 4)

20. +∞, −∞

5.Review 1. 250𝑥11 𝑦12

2. 8𝑥30

𝑦15

3. 𝑟7𝑡3 4. 12𝑥3 + 2𝑥2 + 12𝑥 − 10 5. 𝑥4 + 5𝑥3 − 6𝑥2 + 20𝑥 − 40

6. 10𝑥 − 53 +277𝑥−57

𝑥2+5𝑥−1

7. 10𝑥2 − 43𝑥 + 174 +−700

𝑥+4

8. (2𝑥 − 1)(𝑥2 − 2) 9. (3𝑥2 + 1)(𝑥 − 2)(𝑥 + 2) 10. 3(𝑥 + 2)(𝑥2 − 2𝑥 + 4) 11. -3, -1, 2 12.

±1

4, ±

1

2, ±

3

4, ±1, ±

3

2, ±

9

4, ±3, ±

9

2, ±9

13. −4, 1, 3 14. −1, 1 ± 𝑖 15. 𝑥3 − 5𝑥2 + 8𝑥 − 6

16.

17.

18. 19.x-intercepts: -1, 2; max: (-1, 0); min: (1, -4) x -3 -2 -1 0 1 2 3 4 y -14 -1 2 1 2 11 34 77 \/ \/ \/ \/ \/ \/ \/ 1st 13 3 -1 1 9 23 43 \/ \/ \/ \/ \/ \/ 2nd -10 -4 2 8 14 20 \/ \/ \/ \/ \/ 3rd 6 6 6 6 6 20. 5 units

algebra 2 5 polynomials and polynomial functions practice …rwright/algebra2/homework/chapter...

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