## Question:

Given a polynomial p(x), the value of p(3) = -2.

Which of the following must be true for p(x)?

(A) x - 5 is a factor of p(x).

(B) x - 2 is a factor of p(x).

(C) x + 2 is a factor of p(x).

(D) The remainder when p(x) is divided by x - 3 is -2.

Which of the following must be true for p(x)?

(A) x - 5 is a factor of p(x).

(B) x - 2 is a factor of p(x).

(C) x + 2 is a factor of p(x).

(D) The remainder when p(x) is divided by x - 3 is -2.

## Answer:

If the polynomial p(x) is divided by x−3, the result can be written a

\[ \frac{p(x)}{x-3} = q(x) + \frac{r}{x-3}\]

where q(x) is a polynomial and r is the remainder. Since x−3 is a degree 1 polynomial, the remainder is a real number.

Therefore, p(x) can be rewritten as

In the question it said that p(3) = −2 so it must be true that

−2 = (3−3)q(3) + r =(0)q(3)+r =r

or, r = -2

Therefore, the remainder when p(x) is divided by x−3 is −2.

Therefore, p(x) can be rewritten as

**p(x) = (x−3)q(x) + r**, where r is a real number.In the question it said that p(3) = −2 so it must be true that

−2 = (3−3)q(3) + r =(0)q(3)+r =r

or, r = -2

Therefore, the remainder when p(x) is divided by x−3 is −2.

**Answer is D.**