What is Remainder Theorem? Formula with Example

The Remainder Theorem is an important concept in algebra that helps us quickly find the remainder when a polynomial is divided by a binomial of the form (x − a). Instead of performing full polynomial long division, the theorem provides a simple substitution method to calculate the remainder.

The Remainder Theorem is widely used in solving polynomial equations, checking factors, and simplifying algebraic expressions.

Remainder Theorem Definition

In simple words, the Remainder Theorem states:

If a polynomial f(x) is divided by (x − a), then the remainder is equal to f(a).

This means you can find the remainder just by substituting the value a into the polynomial.

Remainder Theorem Formula

If a polynomial f(x) is divided by (x − a), then:

Remainder = f(a)

Where:

• f(x) = polynomial
• (x − a) = divisor
• a = value obtained by setting the divisor equal to zero
• f(a) = remainder

Example of Remainder Theorem

Find the remainder when:

f(x) = x² + 3x + 2

is divided by:

(x − 1)

Step 1: Identify the Value of a

Since the divisor is (x − 1), we set:

x − 1 = 0
x = 1

So, a = 1

Step 2: Substitute a into the Polynomial

f(1) = (1)² + 3(1) + 2

f(1) = 1 + 3 + 2

f(1) = 6

Final Answer

The remainder is 6.

So, when x² + 3x + 2 is divided by (x − 1), the remainder is 6.

The Remainder Theorem makes polynomial division much faster because it replaces long division with simple substitution.