Answer: x = {-4, -1, 2}
Step-by-step explanation:
q p
F(x) = x³ + 3x² - 6x - 8
[tex]\frac{p}{q} = +/- \frac{1, 2, 4, 8}{1}[/tex]
Possible rational roots are: +/- {1, 2, 4, 8}
F(1) = (1)³ + 3(1)² - 6(1) - 8
= 1 + 3 - 6 - 8
= -10 Since the remainder is not 0, then x = 1 is not a root
F(-1) = (-1)³ + 3(-1)² - 6(-1) - 8
= -1 + 3 + 6 - 8
= 0 Since the remainder is 0, x = -1 is a root.
Use synthetic division to find the remaining factor:
x = -1 → x + 1 = 0
-1 | 1 3 -6 -8
| ↓ -1 -2 8
1 2 -8 0
(x + 1)(x² + 2x - 8) = 0
Next, factor the polynomial:
(x + 1)(x + 4)(x - 2) = 0
x + 1 = 0 x + 4 = 0 x - 2 = 0
x = -1 x = -4 x = 2
