A polynomial p(x) has a remainder of 3 when divided by x - 5. Which of the following equat… — SAT

A polynomial $p(x)$ has a remainder of $3$ when divided by $x - 5$ . Which of the following equations must be true?

The equation

p(5) = 3

must be true.

According to the Remainder Theorem, when a polynomial

p(x)

is divided by a linear expression of the form

x - c

, the remainder is equal to

p(c)

. Here, the divisor is

x - 5

, which gives

c = 5

. The remainder is given as

3

. Therefore, evaluating the polynomial at

x = 5

must yield

3

, which is written as

p(5) = 3

Identify the divisor and the remainder from the given problem statement.

The divisor is

x - 5

and the remainder is

3

Applying the Remainder Theorem requires identifying the value of

c

in the divisor form

x - c

and the remainder value.

Apply the Remainder Theorem to relate the divisor and the remainder to the polynomial function.

Since the divisor is

x - 5

, we have

c = 5

. The Remainder Theorem states that the remainder when

p(x)

is divided by

x - c

p(c)

. Therefore,

p(5) = 3

This establishes the mathematical relationship directly showing which equation must be true.

The Remainder Theorem

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