Question

P(x) is a polynomial with integer coefficients and p(-3) = 0.

Which statements must be true? Choose all that apply.
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 the polynomial.

Explanation:

We have been given that p(x) is a polynomial with integer coefficient.

Also p(-3)=0

Since, p(-3) =0, hence, we can say that -3 is a zero of the polynomial.

Now, we apply factor theorem.

Factor Theorem: If 'a' is a zero of a function f(x) then (x-a) must be a factor of the function f(x).

Applying this theorem, we can say that (x+3) must be a factor of the polynomial.

Hence, first statement must be true.

Answer 2

Answer: The correct option is (A) (x + 3) is a factor of the polynomial.

Step-by-step explanation: Given that p(x) is a polynomial with integer coefficients and p(-3)=0.

We are to select the true statement from the given options.

Factor Theorem: If q(x) is a polynomial with integer coefficients and q(a) = 0, then (q - a) will be a factor of q(x).

Here, it is given that

p(x) is a polynomial with integer coefficients and p(-3) = 0.

Therefore, by Factor theorem, we can say that (x-(-3)), ie., (x + 3) is a factor of the polynomial p(x).

Thus, (x + 3) is a factor of the polynomial.

Option (A) is CORRECT.