Question

P(x) is a polynomial with integer coefficients and p(-3) = 0. Which statements must be true? Choose all that apply. (MORE THAN ONE ANSWER)

x + 3 is a factor of the polynomial.
x - 3 is a factor of the polynomial.
-3 is the constant term of the polynomial.
p(x) can have at most 3 linear factors.

Answer 1

x+3 is a factor of P

Explanation:

By the factor theorem:

P(a)=0 <-> x-a is a factor of P.

You have P(-3)=0.

This implies x-(-3) is a factor of P.

Note: x-(-3)=x+3.

Answer 2

Your answer is the first option, (x + 3) is a factor of the polynomial.

We know this from the factor theorem, which states that if f(a) = 0, then (x - a) is a factor.
Thus we know that (x - -3) is a factor, which simplifies to (x + 3).

All the other answer options do not have to be true, because (x - 3) doesn’t have to be factor, -3 doesn’t have to be the constant of the polynomial, and p(x) can have any number of linear factors, which means the only answer you should select as definitely true is the first option.

I hope this helps! Let me know if you have any questions :)