If g(x) is a polynominal function and g(5) =12, then is x-5 a factor of g(x)? Explain your reasoning.

Question

If g(x) is a polynominal function and g(5) =12, then is x-5 a factor of g(x)? Explain your reasoning.

Answer 1

This is not a true statement. If (x - 5) is a factor of g(x) then g(x) = (x - 5)•Q(x) where Q(x) is the quotient. If we do g(5) we will get:

(5 - 5)•Q(5) and since 5 - 5 = zero, g(5) = zero not 12

Answer 2

No. x-5 would not be a factor of g(x). Since g(5) ≠ 0. (x-5) cannot be a factor.

For example: f(x) = (x + 3)(x+6)

f(-3) = 0 : thus x+3 is a factor of f(x)

f(-2) = 1(4) = 4 ≠ 0 : thus x+2 is not a factor of f(x)

In general, for (x - A) to be a factor of f(x) for any number A, f(A) must be equal to 0.