What is Remainder Theorem? Formula with Example

The Remainder Theorem gives a shortcut for polynomial division. Instead of doing a full long division, you can find the remainder by substituting one value into the polynomial.

Remainder Theorem definition

If a polynomial f(x) is divided by (x − a), the remainder is f(a). In plain words, set the divisor equal to zero to find a, then put that value into the polynomial.

Formula

Remainder = f(a), where f(x) is the polynomial, (x − a) is the divisor, and a is the value that makes the divisor zero.

Worked example

Find the remainder when f(x) = x² + 3x + 2 is divided by (x − 1).

Step 1: x − 1 = 0, so x = 1, and a = 1.

Step 2: f(1) = (1)² + 3(1) + 2 = 1 + 3 + 2 = 6.

Answer: The remainder is 6.

Why it helps

It replaces long division with one substitution, which is useful when checking factors and solving polynomial equations. If f(a) = 0, then (x − a) is a factor of the polynomial.

Learn more

For a video explanation of this topic, see Khan Academy Algebra course.