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1. Describe and correct the error in the following statement.

If 26 is a solution of the polynomial equation f (x) 5 0, then 26 is a factor of f (x).

State the maximum number of turns in the graph of the function.

2. f (x) 5 x4 1 2x2 1 4 3. f (x) 5 23x3 1 x2 2 x 1 5 4. g(x) 5 2x6 1 1

5. g(x) 5 4x2 2 5x 1 3 6. h(x) 5 3x7 2 6x2 1 7 7. h(x) 5 2x9 2 8x7 1 7x5

Determine the x-intercepts of the function.

8. g(x) 5 (x 1 3)(x 2 2)(x 2 5) 9. h(x) 5 (x 1 4)(x 2 6)(x 2 8)

10. f (x) 5 (x 1 3)2(x 2 2) 11. f (x) 5 (x 1 5)(x 1 1)(x 2 7)

12. g(x) 5 (x 1 6)3(x 1 2) 13. h(x) 5 (x 2 8)5

Graph the function.

14. f (x) 5 (x 2 3)(x 1 2)(x 1 1) 15. g(x) 5 (x 2 3)2(x 1 2)

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2

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16. h(x) 5 0.3(x 1 6)(x 2 1)(x 2 4) 17. g(x) 5 5 }

6 (x 1 1)2(x 2 1)(x 2 4)

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2

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18. h(x) 5 (x 2 1)(x2 1 x 1 1) 19. f (x) 5 (x 1 2)(x2 1 2x 1 2)

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1

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1

Practice BFor use with the lesson “Analyze Graphs of Polynomial Functions”

Algebra 2Chapter Resource Book 2-89

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Practice B continuedFor use with the lesson “Analyze Graphs of Polynomial Functions”

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.

20.

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1

21.

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22.

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Use a graphing calculator to graph the function. Identify the x-intercepts and points where local maximums or local minimums occur.

23. f (x) 5 3x3 2 9x 1 1 24. h(x) 5 2 1 } 3 x3 1 x 2

2 }

3

25. g(x) 5 2 1 } 4 x4 1 2x2 26. f (x) 5 x5 2 6x3 1 9x

27. h(x) 5 x5 2 5x3 1 4x 28. g(x) 5 x4 2 2x3 2 3x2 1 5x 1 2

29. Food The average number E of eggs eaten per person

0 10 205 15 25 30 t0

200225250275300325

E

Years since 1970

Per

cap

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egg

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Per Capita EggConsumption each year in the United States from 1970 to 2000 can be

modeled by

E 5 0.000944t4 2 0.052t3 1 0.95t2 2 9.4t 1 308

where t is the number of years since 1970. Graph the function and identify any turning points on the interval 0 ≤ t ≤ 30. What real-life meaning do these points have?

30. Quonset Huts A Quonset hut is a dwelling shaped like half a cylinder. You have 600 square feet of material with which to build a Quonset hut.

a. The formula for surface area is S 5 πr2 1 πrl where r is the radius of the semicircle and l is the length of the hut. Substitute 600 for S and solve for l.

b. The formula for the volume of the hut is V 5 1 }

2  πr2l. Write an equation for the

volume V of the Quonset hut as a polynomial function of r by substituting the expression for l from part (a) into the volume formula.

c. Use the function from part (b) to find the maximum volume of a Quonset hut with a surface area of 600 square feet. What are the hut’s dimensions?

Algebra 2Chapter Resource Book2-90

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Lesson Apply the Fundamental Theorem of Algebra, continuedb. The sum of the zeros of an nth-degree polynomial is equal to the opposite of the coefficient of the xn 2 1 term. c. The product of the zeros of an nth-degree polynomial is equal to the constant term if the function is of even degree and to the opposite of the constant term if the function is of odd degree. 10. The concentration is the greatest after about 4.5 hours.

Lesson Analyze Graphs of Polynomial FunctionsTeaching Guide

1. (0, 3) and (2, 21) 2. (1, 29)

3. (21.7, 10.4) and (1.7, 210.4)

Graphing Calculator Activity

1. local maximum: (0.37, 13.51); local minima: (21.83, 212.66), (2.96, 227.04)

2. local maximum: (22.54, 0.88); local minimum: (20.13, 26.06) 3. local maximum: (0, 36); local minima: (22.55, 26.25), (2.55, 26.25) 4. local maximum (22.67, 29.63); local minimum: (2.0, 272) 5. local maxima: (21.57, 1.92), (2.25, 8.63); local minimum: (0.07, 24.02) 6. local maximum: (24.19, 16.16); local minimum: (2.86, 254.08)

Practice Level A

1. False. Sample answer: If k is an imaginary zero, then it is not an x-intercept of the graph of f (x). 2. 4 3. 5 4. 3 5. (21.5, 5.5): max, (0.5. 3): min 6. (1, 2): min 7. (21.5, 23.75): min, (0.5, 1.25): max, (1, 1): min 8. B 9. C

10. A 11. 24; 1 12. 2; 3 13. 24; 0; 5

14. 23, 21, 8 15. 26 16. 1; 7

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23. a. 4 b. 4 c. (22, 0), (6, 0) 24. a. 0 < x < 9, because the sides must be less than 9 inches. b. 3 in. c. 432 in.3

Practice Level B

1. The error is in calling 26 a factor. The correct statement is If 26 is a solution of the polynomial equation f(x) 5 0, then (x 1 6 ) is a factor of f(x).

2. 3 3. 2 4. 5 5. 1 6. 6 7. 8 8. 23, 2, 5

9. 24, 6, 8 10. 23, 2 11. 25, 21, 7

12. 26, 22 13. 8

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20. (20.5, 0.5): max, (0.5, 20.3): min; 20.6, 0, 0.6; 3 21. (22, 0): min, (20.5, 5): max, (1, 0): min; 22, 1; 4 22. (22.8, 1.9): max, (0, 0.25): min, (2, 1): max; 23.8, 2.8; 4 23. x-int: 21.79, 0.11, 1.67; max: (21, 7); min: (1, 25)

24. x-int: 22, 1; max: (1, 0); min: 1 21,2 4 } 3 2

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Lesson Analyze Graphs of Polynomial Functions, continued25. x-int: 22.83, 0, 2.83; max: (22, 4), (2, 4); min: (0, 0) 26. x-int: 21.73, 0, 1.73; max: (21.73, 0), (0.77, 4.46); min: (20.77, 24.46), (1.73, 0) 27. x-int: 22, 21, 0, 1, 2; max: (21.64, 3.63), (0.54, 1.42); min: (20.54, 21.42), (1.64, 23.63) 28. x-int: 21.53, 20.35, 1.88, 2; max: (0.61, 3.62); min: (21.05, 23.03), (1.94, 20.03)

29.

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Per

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Per Capita EggConsumption

(25.33, 222.93); Sample answer: From 1970 to 1995, per capita egg consumption decreased to about 223 eggs, then began to increase again.

30. a. l 5 600

} πr 2 r b. V 5 300r 2 0.5πr3

c. 1596 ft3; r < 7.98 ft, l < 15.95 ft

Practice Level C

1. 3 2. 2 3. 4

4.

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5.

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domain: 2` < x < ` domain: 2` < x < `

range: 2` < y < ` range: 2` < y < `

6.

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7.

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domain: 2` < x < ` domain: 2` < x < ` range: y ≥ 24.489 range: y ≤ 0.8

8.

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9.

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domain: 2` < x < ` domain: 2` < x < `

range: 2` < y < ` range: 2` < y < `

10.

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11. x

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24028

domain: 2` < x < ` domain: 2` < x < `

range: y ≥ 222.9 range: 2` < y < `

12. (22, 2.4): max, (0, 21.2): min, (1, 21): max, (2.1, 22): min; 22.5, 21, 2.7; 5 13. (22, 21): max, (0, 22.2): min, (1, 22): max; none; 4

14. (21.5, 22.25): min, (0, 3): max, (2, 23): min, (3.5, 2): max; 22, 21, 1, 3, 4; 5

15. x-int: 20.34, 1.43, 2.1; max: (0.7, 2.44); min: (1.83, 21.55) 16. x-int: 0, 1.2; max: (21.19, 4.09); min: (21.77, 3.8), (0.71, 22.93)

17. x-int: 20.88, 1.08, 1.41; max: (1.26, 0.08); min: (0, 20.88) 18. x-int: 25, 0, 1.2; max: (0.42, 1.38); min: (23.57, 2116.16), (1.2, 0)

19. 21 20. 2, 4 21. 1, 3, 5

22. If n is even, there is a turning point. If n is odd, the graph passes through the x-axis.;

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23. a. h 5 500 2 πr2

} πr

b. V 5 500r 2 1 }

3 πr3

c. about 3613.79 ft3; r < 10.84 ft, h < 2.56 ft

Study Guide

1. ; x-intercepts: (22, 0), (4, 0); maximum: (22, 0), minimum: (2, 216)

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