Practice and Problem-Solving Exercises

A Practice

See Problems 1 and 2.

Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots.

9. x 3 − 4 x + 1 = 0 x cubed , minus 4 x plus 1 equals 0
10. x 3 + 2 x − 9 = 0 x cubed , plus 2 x minus 9 equals 0
11. 2 x 3 − 5 x + 4 = 0 2 x cubed , minus 5 x plus 4 equals 0
12. 3 x 3 + 9 x − 6 = 0 3 x cubed , plus 9 x minus 6 equals 0
13. 4 x 3 + 2 x − 12 = 0 4 x cubed , plus 2 x minus 12 equals 0
14. 6 x 3 + 2 x − 18 = 0 6 x cubed , plus 2 x minus 18 equals 0
15. 7 x 3 − x 2 + 4 x + 10 = 0 7 x cubed , minus , x squared , plus 4 x plus 10 equals 0
16. 8 x 3 + 2 x 2 − 5 x + 1 = 0 8 x cubed , plus 2 , x squared , minus 5 x plus 1 equals 0
17. 10 x 3 − 7 x 2 + x − 10 = 0 10 x cubed , minus , 7 x squared , plus x minus 10 equals 0

See Problem 3.

A polynomial function P(x) with rational coefficients has the given roots. Find two additional roots of P ( x ) = 0 . p open x close equals 0 .

18. − 2 i  and  10 negative 2 i , and , square root of 10
19. 14 − 2  and  − 6 i 14 minus square root of 2 , and , minus 6 i
20. i  and  7 + 8 i i , and , 7 plus 8 i
21. − 3  and  5 − 11 negative square root of 3 , and , 5 , negative square root of 11

See Problem 4.

Write a polynomial function with rational coefficients so that P ( x ) = 0 p open x close equals 0  has the given roots.

22. 7 and 12
23. − 9 negative 9  and − 15 negative 15
24. − 10 i negative 10 i
25. 3 i + 9 3 i plus 9
26. 4 , 16 ,  and  1 + 19 i 4 comma 16 comma , and , 1 plus 19 i
27. 13 i  and  5 + 10 i 13 i , and , 5 plus 10 i
28. 11 − 2 i  and  8 + 13 i 11 minus 2 i , and , 8 plus 13 i
29. 17 − 4 i  and  12 + 5 i 17 minus 4 i , and , 12 plus 5 i

See Problem 5.

What does Descartes’ Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function?

30. P ( x ) = x 2 + 5 x + 6 p open x close equals , x squared , plus 5 x plus 6
31. P ( x ) = 9 x 3 − 4 x 2 + 10 p open x close equals , 9 x cubed , minus , 4 x squared , plus 10
32. P ( x ) = 8 x 3 + 2 x 2 − 14 x + 5 p open x close equals , 8 x cubed , plus , 2 x squared , minus 14 x plus 5

B Apply

Find all rational roots for P ( x ) = 0 . p open x close equals 0 .

33. P ( x ) = 2 x 3 − 5 x 2 + x − 1 p open x close equals , 2 x cubed , minus , 5 x squared , plus x minus 1
34. P ( x ) = 6 x 4 − 13 x 3 + 13 x 2 − 39 x − 15 p open x close equals , 6 x to the fourth , minus , 13 x cubed , plus , 13 x squared , minus 39 x minus 15
35. P ( x ) = 7 x 3 − x 2 − 5 x + 14 p open x close equals , 7 x cubed , minus , x squared , minus 5 x plus 14
36. P ( x ) = 3 x 4 − 7 x 3 + 10 x 2 − x + 12 p open x close equals , 3 x to the fourth , minus , 7 x cubed , plus , 10 x squared , minus x plus 12
37. P ( x ) = 6 x 4 − 7 x 2 − 3 p open x close equals , 6 x to the fourth , minus , 7 x squared , minus 3
38. P ( x ) = 2 x 3 − 3 x 2 − 8 x + 12 p open x close equals , 2 x cubed , minus , 3 x squared , minus 8 x plus 12

Write a polynomial function P ( x ) p open x close  with rational coefficients so that P ( x ) = 0 p open x close equals 0  has the given roots.

39. − 6 , 3 , negative 6 comma 3 comma  and − 15 i negative 15 i
40. 4 + 5  and  8 i 4 plus square root of 5 , and , 8 i
41. − 5 − 7 i  and  2 − 11 negative 5 minus 7 i , and , 2 , negative square root of 11

42. Think About a Plan You are building a square pyramid out of clay and want the height to be 0.5 cm shorter than twice the length of each side of the base. If you have 18 cm 3 18 cm cubed  of clay, what is the greatest height you could use for your pyramid?

    • How can drawing a diagram help you solve this problem?
    • What is the formula for the volume of a pyramid?
    • What equation can you solve to find the height of the pyramid?

43. Error Analysis Your friend is using Descartes’ Rule of Signs to find the number of negative real roots of x 3 + x 2 + x + 1 = 0 . x cubed , plus , x squared , plus x plus 1 equals 0 .  Describe and correct the error.

    
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44. Reasoning A quartic equation with integer coefficients has two real roots and one imaginary root. Explain why the fourth root must be imaginary.

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