Name———————————————————————— Lesson
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 iÏ 5 14. Ï 2 2 iÏ 7 15. 23, 2, 9
}
}
16. 24, 1, 5 17. 26, 2i, i 18. 28, 2Ï 7 , Ï 7 }
}
19. 2, 23i, 3i 20. 2Ï 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 iÏ 3 , 2 2 iÏ 3 4. 24, 22, } 2 , 4 }
9 23 6 iÏ19 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 , 2Ï3 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
