1. The Real Zeros of a Polynomial Function
Section 5.2
The Real Zeros of a Polynomial Function

2. OBJECTIVE 1

5. (a) f(-2) = (-2)3 + 3(-2)2 + 2(-2) -1 = -1
(b) f(1) = (1)3 + 3(1)2 + 2(1) -1 = 5

7. a) f(-1) = -2(-1)3 - (-1)2 + 4(-1) + 3 = 0  x + 1 is a factor
b) f(1) = -2(1)3 – (1)2 + 4(1) + 3 = 4  x - 1 is NOT a factor

9. OBJECTIVE 2

11. p: ±1, ±2, ±3, ±4, ±6, ±12
q: ±1, ±3
p/q: ±1, ±1/3, ±2, ±2/3, ±3, ±4, ±4/3, ±6, ±12

12. OBJECTIVE 3

14. p/q: ±1, ±1/3, ±2, ±2/3, ±3, ±4, ±4/3, ±6, ±12

15. f(-3) = 3(-3) (-3) (-3) – 12 = 0
f(-1) = 3(-1) (-1) (-1) – 12 = 0
f(4/3) = 3(4/3) (4/3)2 - 7(4/3) – 12 = 0
-3 |
____________
(x + 3)(3x2 –x - 4)
(x + 3)(3x – 4)(x + 1)

17. p: ±1, ±3, ±9
q: ±1, ±2
p/q: ±1, ±1/2, ±3, ±3/2, ±9, ±9/2

18. p/q: -1, ±1/2, -3, -3/2, ±9, ±9/2

19. f(-3) = 2(-3)4 + 13(-3)3 + 29(-3)2 + 27(-3) + 9 = 0
-1 |
_________________
(x + 1)(2x x x + 9)
-3 |
_________________
(x + 1)(x + 3)(2x2 + 5x + 3) = (x + 1)(x + 3)(2x + 3)(x + 1)
= (x + 1)2(x + 3)(2x + 3)

20. OBJECTIVE 5

21. BOUND

22. Smaller of two values is 10 Bound is -10 ≤ x ≤ 10
g(x) = 4(x5 – 1/2x3 + 1/2x2 + 1/4)
Max{ 1, } = Max{1, 17}
= 17
Max{ 1, 1/2 + 1/2 + 1/4}
= Max{1, 5/4}
= 5/4
1 + Max{3, 9, 5} = = 10
1 + Max{1/2, 1/2, 1/4}
= 1 + ½ = 3/2
Smaller of two values is 10
Bound is -10 ≤ x ≤ 10
Smaller of two values is
5/4. Bound is -5/4 ≤ x ≤ 5/4

24. May be at most 5 zeros.
Can NOT use Rational Zeros Theorem (non-integer
coefficients)
Find bounds:
Max{1, }
= Max{1, } =
1 + Max{1.8, 17.79, , 37.95, } = 38.95
Bound: 38.95

26. OBJECTIVE 6

28. Using the function from the last example, determine whether there is a repeated zero or two distinct zeros near 3.30.
f(3.29) = > 0 and
f(3.30) = < 0
f(3.30) = < 0 and
f(3.31) = > 0

29. Use xmin 3.29 and xmax 3.31
Use ymin and ymax