If a polynomial p(x) is divided by x - 5, the remainder is 12. Which of the following equa… — SAT

If a polynomial $p(x)$ is divided by $x - 5$ , the remainder is $12$ . Which of the following equations must be true?

p(5) = 12

According to the Remainder Theorem, dividing a polynomial

p(x)

by a linear expression

x - c

results in a remainder equal to

p(c)

. Here, the divisor is

x - 5

, so

c = 5

. Since the remainder is

12

, the equation that must be true is

p(5) = 12

Identify the divisor and its corresponding value of

c

using the Remainder Theorem.

The divisor is

x - 5

, so we set

x - 5 = 0

, which gives

c = 5

The Remainder Theorem states that the remainder of a polynomial

p(x)

divided by

x - c

p(c)

Set

p(c)

equal to the given remainder.

Since the remainder is

12

and

c = 5

, we get

p(5) = 12

This directly applies the relation

p(c) = \text{remainder}

The Remainder Theorem

Question