Question

Given a polynomial f(x), if (x + 3) is a factor, what else must be true?

Answer 1

One of the zeroes must be x=-3.

Answer 2

What applies here is one of the laws of the factorization of polynomials, called the factor theorem and it states that:

For a polynomial f(x) , if for any value a, f(a) =0 then (x-a) is factor of f(x)

Example:

Consider the polynomial

`f(x) = x^(3) - 3x^(2) - 8x+4`

For a =3,

`f(a) = (3)^(3) - 3(3)^(2) - 8(3)+24`

= 27-27-24+24 = 0

f (3)= 0

which means (x-3) is a factor of f(x)

Applying the above rule to the question:

if (x + 3) is a factor of a polynomial f(x), then f(-3) = 0

note that (x+3) can also be written as (x- (-3)).