Remainder Theorem

The remainder theorem is a fundamental concept in algebra that allows you to quickly find the remainder of a polynomial when divided by a linear expression. By using this formula, it helps easily find the remainder without having to do the long division. In this guide, we will learn the remainder theorem, its statement, and related examples to build a strong understanding of the concept.

Table of Contents

• What is the Remainder Theorem
• How to Find Remainder Using the Remainder Theorem
• Solved Examples on Remainder Theorem

What is the Remainder Theorem

Statement: If a polynomial p(x) is divided by a linear polynomial (x - a), then the remainder is p(a).

Proof:

Let p(x) be a polynomial and (x - a) be a linear expression. Let q (x) and r be the quotient and remainder, respectively, when p(x) is divided by (x - a)

p(x) = q(x)(x - a) + r
Substitute x = a
p(a) = q(a)(a - a) + r
⇒ p(a) = r
i.e., the remainder is equal to p(a).

NOTE: The degree of the remainder polynomial is always 1 less than the degree of the divisor polynomial. ∴ When any polynomial is divided by a linear polynomial (whose degree is 1), the remainder must be a constant (whose degree is 0).

How to Find Remainder Using the Remainder Theorem

To find the remainder using the remainder theorem, follow the steps below:

• Find the zero of the linear polynomial (x - a) = 0, which is x = a.

• Substitute the obtained zero in the given polynomial p (x). The value obtained, p(a), is the required remainder.

• When dividing by (ax + b): remainder = p (-b/a)
  
  

Solved Examples on Remainder Theorem

Example 1: Find the remainder when p(x) = x³ − 5x² + 4x − 12 is divided by (x − 2).
Solution: Given p(x) = x³ − 5x² + 4x − 12 and x = 2.
p (2) =  2³ − 5(2²) + 4(2) − 12 = 8 − 20 + 8 − 12 =  -16
∴ the remainder when p(x) = x³ − 5x² + 4x − 12 is divided by (x − 2) is −16.

Example 2: Find the remainder when p(x) = 2x³ + x² − 5x + 1 is divided by (x + 3).
Solution: Given p(x) = 2x³ + x² − 5x + 1 and x = -3.
p (-3) =   2(-3)³ + (-3)² − 5(-3) + 1 = -54 + 9 + 15 + 1 = -29
∴ the remainder when p(x) = 2x³ + x² − 5x + 1 is divided by (x + 3) is −29.

Example 3: Find the remainder when p(x) = 4x³ − 12x² + 11x − 3 is divided by (2x + 1).
Solution: Given p(x) = 4x³ − 12x² + 11x − 3 and x = -1/2,
p (−1/2) = 4(-½)³ − 12(-½)² + 11(-½) − 3 = -1.
∴ the remainder when p(x) = 4x³ − 12x² + 11x − 3 is divided by (2x + 1) is −29.