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EXERCISES
For more practice, see Extra Practice.
Practice and Problem Solving
A
Practice by Example
Use the Rational Root Theorem to list all possible rational roots for each
polynomial equation. Then find any actual rational roots.
Example 1
1. x 3 - x 2 + 2x - 2 = 0
2. x 3 + 4x 2 + x - 6 = 0
(page 330)
3. x 3 + x 2 + 4x + 4 = 0
4. 2x 3 - 9x 2 - 11x + 8 = 0
5. x 3 + 2x 2 - 8x - 16 = 0
6. x 4 + 2x 2 - 15 = 0
Example 2
(pages 330–331)
Find the roots of each polynomial equation.
7. x 3 - 2x 2 + 5x - 10 = 0
9. 2x 4 - 5x 3 - 17x 2 + 41x - 21 = 0
11. 4x 4 - 37x 2 + 9 = 0
Examples 3 and 4
(pages 331 and 332)
Example 5
(page 332)
B
Apply Your Skills
8. x 3 - 5x 2 + 7x - 35 = 0
10. 4x 3 + 16x 2 - 22x - 10 = 0
12. 9x 4 + 3x 3 - 30x 2 + 6x + 12 = 0
A polynomial equation with rational coefficients has the given roots. Find two
additional roots.
13. !5 and 2!13
14. 4 - !6 and !3
15. 1 - !10 and 2 + !2
16. 1 + i and -5i
17. 2 + 3i and 6i
18. 4 - i and 3 + 7i
Find a third-degree polynomial equation with rational coefficients that has the
given numbers as roots.
19. 1 and 3i
20. -5 and 1 - i
21. 2 and -4i
22. 3 + i and -3
23. -2i and 6
24. -1 and i + 1
Use the Rational Root Theorem to list all possible rational roots for each
polynomial equation. Then find any actual rational roots.
25. 12x 3 - 32x 2 + 25x - 6 = 0
26. 10x 3 - 49x 2 + 68x - 20 = 0
27. 6x 4 - 5x 3 - 65x 2 + 85x - 21 = 0
28. 8x 3 - 28x 2 + 14x + 15 = 0
Find a fourth-degree polynomial equation with integer coefficients that has the
given numbers as roots.
29. 3 + i and -2i
30. !3 and 1 - i
31. 3 + !2 and !5
In each equation, r, s, and t represent integers. Indicate whether the statement is
sometimes, always, or never true. Explain your answer.
32. A root of the equation 3x 3 + rx 2 + sx + 8 = 0 is 5.
33. A root of the equation 3x 3 + rx 2 + sx + 8 = 0 is -2.
34. If a is a root of x 3 + rx 2 + sx + t = 0, then a is a factor of t.
35. !5 and 2!5 are roots of x 3 + rx 2 + sx + t = 0.
36. 2 + i and -2 - i are roots of x 3 + rx 2 + sx + t = 0.
37. Error Analysis A student claims that 2i is the only imaginary root of a
polynomial equation that has real coefficients. Explain the student’s mistake.
38. Open-Ended Write a fourth-degree polynomial equation with integer
coefficients that has two irrational roots and two imaginary roots.
Lesson 6-5 Theorems About Roots of Polynomial Equations
333-334
C
39. Critical Thinking Explain why the Irrational Root Theorem requires that !b
of a + !b be an irrational number.
Challenge
40. a. Using real and imaginary as types of roots, list all possible combinations of
root type for a fourth-degree polynomial equation.
b. Repeat the process for a fifth-degree polynomial equation.
c. Make a Conjecture Make a conjecture about the number of real roots of an
odd-degree polynomial equation.
41. Writing A student states that 2 + !3 is a root of x 2 - 2x - (3 + 2!3) = 0.
The student claims that 2 - !3 is another root of the equation by the
Irrational Root Theorem. Explain how you would respond to the student.
42. What polynomial equation with complex coefficients and no multiple roots has
-4i and 2 + 3i as its only roots?
43. a. Find a polynomial equation in which 1 + !2 is the only root.
b. Find a polynomial equation with root 1 + !2 of multiplicity 2.
c. Find c such that 1 + !2 is a solution of x 2 - 2x + c = 0.
Standardized Test Prep
Multiple Choice
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44. Three roots of a polynomial equation with rational coefficients are 5 + !3,
-17, and 2 - !4. Which number also is a root of the equation?
A. 17
B. 2 + !4
C. 4 - !2
D. 5 - !3
45. Two roots of a cubic polynomial equation with real coefficients are -3 and
-4i. If the leading coefficient of the polynomial is 1, what is the equation?
G. x 3 - 3x 2 - 16x + 48 = 0
F. x 3 - 3x 2 + 16x - 48 = 0
H. x 3 + 3x 2 + 16x + 48 = 0
I. x 3 + 3x 2 - 16x - 48 = 0
Short Response
46. According to the Rational Root Theorem, what is the relationship between
the polynomial equation 2x 4 - x 3 - 7x 2 + 3x + 3 = 0 and rational roots
p
p
of the form q, where q is in simplest form?
Extended Response
47. A third-degree polynomial equation with rational coefficients has roots -4
and -4i. If the leading coefficient of the equation is 32, what is the
equation? Show your work.
Mixed Review
Lesson 6-4
Solve each equation.
48. 8x 3 + 27 = 0
Lesson 5-7
52. p 2 + 4p = -8
53. 4x 2 - 11 = 12x
Use Cramer’s Rule to solve each system.
54. e
333-334
50. 2x 4 - 50 = 0
Solve each equation by completing the square.
51. x 2 - 6x - 7 = 0
Lesson 4-8
49. x 4 - x 2 - 20 = 0
23x 1 2y 5 27
25x 1 2y 5 23
Chapter 6 Polynomials and Polynomial Functions
55. e
2x 2 3y 5 212
2x 1 7y 5 22
56. e
22x 2 8y 5 210
23x 1 8y 5 215