10. Let g(x) = (x - a)⁵ - x⁵ + a⁵. Show that f(x) is divisible by x and by x - a, and find the other factors.
 
11. Find all the integral roots of 3x⁴ + 20x³ + 36x² + 16x = 0, and then find the other roots.
 
12. Let f(x) = a₀xⁿ + a₁x^{n -1} + ... + a_{n -1}x; that is, let a_n = 0. Also, let the a_i be integers. Show that any non-zero integral root of f(x) = 0 is an integral divisor of a_{n -1}.
 
13. Find a rational root of 3x³ + 4x² - 21x + 10 = 0, and then find the other roots.
 
14. Find all the roots of 6x⁴ + 31x³ + 25x² - 33x + 7 = 0.
 
15. Find all the roots of 81x⁵ - 54x⁴ + 3x² - 2x = 0.
 
16. Let f(x) = a₀xⁿ + a₁x^{n -1} + ... + a_{n -1}x with the a_i integers. State a necessary condition for a non-zero rational number to be a root of f(x) = 0.
 
17. Given that a and b are integers, what are possibilities for rational roots of
x³ + ax² + bx + 30 = 0?
 
18. Let f(x) = xⁿ + a₁x^{n -1} + ... + a_{n -1}x + a_n with the a_i integers. Note that a₀ = 1. Show that any rational root of f(x) = 0 must be an integer.
 
19. Let f(x) be a polynomial. Let r and s be roots of f(x) = 0 and let Show that there exist polynomials g(x) and h(x) such that all of the following are true:
 
(a) f(x) = (x - r)g(x).
 
(b) g(s) = 0.
 
(c) g(x) = (x - s)h(x).
 
(d) f(x) = (x - r)(x - s)h(x).
 
20. Let f(x) = 0 be a polynomial equation with distinct roots r, s, and t. Show that
f(x) = (x - r)(x - s)(x - t)p(x), with p(x) a polynomial.