Practice B

Find all the zeros of the polynomial function. 5. h(x) 5 x3 2 3x2 2 x 1 3. 6. f(x) 5 x4 2 ... revenue $1.5 million? Practice B. For use with the lesso...

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2.7

Date —————————————

Practice B

For use with the lesson “Apply the Fundamental Theorem of Algebra”

Identify the number of solutions or zeros. 1. f(x) 5 5x 3 2 6x 2 1 2x 2 3

2. g(s) 5 8s 6 2 3s 4 2 11s 3 2 2s 2 1 4

3. 23y 7 1 5y 5 2 12y 1 2 5 6

4. 4 2 7x 5 x 2 2 3x 5

Find all the zeros of the polynomial function.

5. h(x) 5 x 3 2 3x 2 2 x 1 3

6. f (x) 5 x 4 2 4x 3 2 20x 2 1 48x

7. g(x) 5 x 3 1 5x 2 1 x 1 5

8. g(x) 5 x 4 2 9x 3 1 23x 2 2 81x 1 126

9. f(x) 5 x 3 2 x 2 2 11x 1 3

10. h(x) 5 2x 4 1 x 3 1 x 2 1 x 2 1

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

11. 27, 24

12. 1, 2, 5

13. 23, 0, 1

14. 4, i, 2i

15. 25, 0, 22i, 2i

16. 8, 2 1 i

17. Multiple Choice  Which is not a possible classification of the zeros of

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

18. g(x) 5 x 3 2 x 2 2 5x 1 3

19. h(x) 5 2x 3 2 x 2 2 3x 2 1

20. f(x) 5 x 4 2 2x 2 1

21. g(x) 5 x 4 2 x 3 2 20x 2 1 10x 1 27

22. Sporting Goods  For 1998 through 2005, the sales S (in billions of dollars) of sporting goods can be modeled by

S 5 0.007t 3 1 0.1t 2 1 1.4x 1 70

where t is the number of years since 1998. In which year were sales about $78 billion?

Lesson 2.7

Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.



f(x) 5 x 4 1 2x 3 2 7x 2 2 7x 1 3 according to Descartes’ rule of signs? A. 2 positive real zeros, 2 negative real zeros, and 0 imaginary zeros B. 0 positive real zeros, 2 negative real zeros, and 2 imaginary zeros C. 0 positive real zeros, 0 negative real zeros, and 4 imaginary zeros D. 1 positive real zero, 1 negative real zero, and 2 imaginary zeros

23. Grocery Store Revenue  For the 25 years that a grocery store has been open, its annual revenue R (in millions of dollars) can be modeled by

1

     ​ (2t 4 1 12t 3 2 77t 2 1 600t 1 13, 650) R 5 ​ } 10,000

where t is the number of years the store has been open. In what year(s) was the revenue $1.5 million? Algebra 2 Chapter Resource Book

2-75

Lesson 2.6 Find Rational Zeros, continued b.

25

24

23

22

21

f(x)

299

247

217

23

1

1. 2 2. 3 3. 3 4. 4 5. 5 6. 4 7. 1 8. 3 }

}

9. 3 1 ​Ï 2 ​  10. 2i 11. 22 1 9i 12. ​Ï 3 ​ 1 i }

}

}

13. 1 2 i​Ï 5 ​  14. Ï ​ 2 ​ 2 i​Ï 7 ​  15. 23, 2, 9

}

}

16. 24, 1, 5 17. 26, 2i, i 18. 28, 2​Ï 7 ​,  ​Ï 7 ​  }

}

19. 2, 23i, 3i 20. 2​Ï 5 ​,  ​Ï 5 ​,  22i, 2i

x

0

1

2

3

4

5

f(x)

1

3

13

37

81

151

21. f (x) 5 x 1 9 22. f (x) 5 x 2 1 x 2 20 23. f (x) 5 x 2 1 4x 1 3 24. f (x) 5 x 3 2 x 25. f (x) 5 x 3 2 7x 2 1 4x 1 12

The polynomial is guaranteed to have a zero in 22 ≤ x ≤ 21.

26. f (x) 5 x3 2 7x2 2 28x 2 20

c.

28. about 0.28 in. or about 3.70 in.

x

22

21.9

21.8

21.7

27. 2 real zeros; no imaginary zeros

Practice Level B

21.6

f(x) 23 22.249 21.592 21.023 20.536 x

21.5

21.4 21.3 21.2 21.1 21

Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.

f(x) 20.125 0.216 0.493 0.712 0.879

1

The polynomial is guaranteed to have a zero in 21.5 ≤ x ≤ 21.4. d. The zero approximated to three decimal places is 21.446. 3. The optimal order size is between 4000 and 5000 units.

1. 3 2. 6 3. 7 4. 5 5. 21, 1, 3 6. 24, 0, 2, 6 7. 25, 2i, i 8. 2, 7, 23i, 3i } } 1 9. 23, 2 1 Ï ​ 3 ​ , 2 2 Ï ​ 3 ​  10. 21, } ​ 2 ​, 2i, i

11. f (x) 5 x 2 1 11x 1 28 12. f (x) 5 x 3 2 8x 2 1 17x 2 10 13. f (x) 5 x 3 1 2x 2 2 3x 14. f (x) 5 x 3 2 4x 2 1 x 2 4 15. f (x) 5 x 4 1 5x 3 1 4x 2 1 20x 16. f (x) 5 x 3 2 12x 2 1 37x 2 40 17. D 18. 22.09, 0.57, 2.51 19. 20.62, 20.50, 1.62 20. 20.47, 1.40 21. 24.09, 20.98, 1.47, 4.60

Lesson 2.7 Apply the Fundamental Theorem of Algebra

22. 2002 23. 3 and 9 years after opening

Practice Level C

Teaching Guide

}

1. If f(x) 5 anx 1 . . . 1 a1x 1 a0 has integer coefficients, then every rational zero of f  has the n

p

answers

x

Practice Level A

factor of constant term a0

​ factor of leading         ​. following form: } ​ q ​5 }}} coefficient a n

1 2. 61, 62, 64, 68, 616, 6​ } ​ 2 } 2 2b 6 Ï ​ b 2 4ac ​  3. }} ​       ​;  two imaginary solutions 2a 4. two complex numbers of the form a 1 bi and

}

1. 23, 21, 3 2. 21, 1, 2 1 Ï ​ 3 ​,  2 2 ​Ï3 ​  } } 5 3. 22, 0, 2 1 i​Ï 3 ​,  2 2 i​Ï 3 ​  4. 24, 22, } ​ 2 ​, 4 }

9 23 6 i​Ï19 ​  5. 23, 21, 3, } ​ 2 ​ 6. 1, ​ }     ​  2 }

}

7. 2i, i, 21 1 Ï ​ 2 ​,  21 2 Ï ​ 2 ​  8. 1 1 4i, 1 2 4i, ​ } } Ï3 ​,  2​Ï3 ​  9. f (x) 5 2x 4 2 4x 3 2 32x 2 1 64x 10. f (x) 5 2x 4 1 58x 2 1 200 11. f (x) 5 2x 3 1 10x 2 2 6x 2 30

a 2 bi; 31

12. f (x) 5 2x 3 2 12x 2 1 50x

Investigating Algebra Activity

13. f (x) 5 2x 4 2 22x 3 1 80x 2 2 116x 1 56

1. a. 2, 2 b. 3, 3 c. 4, 4 2. They are the same. 3. Sample answer: The degree of a polynomial

indicates how many zeros, or solutions, the polynomial will have. For example, x 5 2 3x 2 1 32 is a polynomial of degree 5, so it will have 5 solutions.

14. f (x) 5 2x 6 2 26x 5 1 120x 4 2 200x 3 1 118x 2 2 174x 15. 2 or 0; 1; 2 or 0 16. 2 or 0; 0; 4, 2, or 0 17. 4, 2, or 0; 1; 4, 2, or 0 18. 5, 3, or 1; 5, 3, or 1; 8, 6, 4, 2, or 0 19. 3 or 1; 3 or 1; 2 or 0 20. 20.75, 0.75 21. 20.86 22. 21.06, 3.98

Algebra 2 Chapter Resource Book

A29