Dividing polynomials

Question

F(X)=g(X)+2x+2

F(X) is divisible by x² -5x-6

g(X) is divisible by x² -2x-3

f(x) and g(x) are unknown

determine the values of f(6) and g(6) and determine the common factor for the two functions

Answer 1

Since f(x) is divisible by x² - 5x - 6, f(x) = P(x)(x² - 5x - 6) = P(x)(x - 6)(x + 1), for some polynomial function P(x).

So, f(6) = P(6)(0)(7) = 0.

Since g(x) is divisible by x² - 2x - 3, g(x) = Q(x)(x² - 2x - 3) = Q(x)(x-3)(x+1), for some polynomial function Q(x).

We see that f(x) and g(x) have x + 1 as a common factor.

Since f(x) = g(x) + 2x + 2, f(6) = g(6) + 14

0 = g(6) + 14

g(6) = -14