Problems for Sections 8.1, 8.2, and 8.3

1. Express p(x) = x⁴ + 5x³ - 10x - 12 in the form (x + 2)q(x) + p(-2).

2. Express f(x) = 5x⁵ - x⁴ - x³ - x² - x - 2 in the form (x - 1)g(x) + f(1).

3. Show that -1 is a root of x³ + 3x² - 2 = 0, and find the other roots.

4. Show that 2 is a root of x³ - 6x + 4 = 0, and find the other roots.

5. Find a, given that -4 is a root of 5x⁶ - 7x⁵ + 11x + a = 0.

6. Find b, given that 3 is a root of x⁷ - 10x⁵ + 8x³ + 4x² - 3x + b = 0.

7. Find all the integral roots of x⁴ - 2x³ - x² - 4x - 6 = 0, and then find the other roots.

8. Find all the integral roots of x⁵ - 8x⁴ + 15x³ + 8x² - 64x + 120 = 0, and then find the other roots.

9. Let f(x) = (x - a)³ - x³ + a³. Find f(0) and f(a), and use this information to find two factors of f(x).