Question

Given a polynomial f(x), if (x + 5) is a factor, what else must be true?

A. f(0) = 5
B. f(0) = −5
C. f(5) = 0
D. f(−5) = 0

Answer 1

Given a polynomial f(x), if (x + 5) is a factor, D. f(−5) = 0 must be true

Step-by-step explanation:

If (x + 5) is a factor of the polynomial f(x), then the statement "f(−5) = 0" must be true. This is because when a factor is (x - c), where c is a root of the polynomial, substituting x = c into the polynomial will make the factor equal to zero. In this case, since (x + 5) is a factor, setting x = -5 makes f(−5) equal to zero.

When (x + 5) is a factor, it implies that the polynomial has a root at x = -5. Therefore, plugging x = -5 into the polynomial will result in f(−5) = 0. This is a fundamental property of polynomial factors and roots – when a factor is equal to zero, it corresponds to a root of the polynomial.

In summary, the correct statement is D. f(−5) = 0, reflecting the fact that (x + 5) being a factor implies that x = -5 is a root of the polynomial, making the polynomial evaluate to zero at that point.