Remainder Theorem

Lesson Objectives

By the end of this lesson, students should be able to:

State the Remainder Theorem.
Apply the Remainder Theorem to evaluate remainders.
Determine if a number is a root of a polynomial using substitution.
Relate Remainder Theorem to Factor Theorem conceptually.

Introduction

Imagine you have a machine that divides polynomials and instantly tells you the remainder. That’s exactly what the Remainder Theorem does! Instead of doing long division, you can simply substitute a number into the polynomial. Let’s learn how!

Core Lesson Content

The Remainder Theorem states that: If a polynomial $f(x)$ is divided by $x - a$ , the remainder is $f(a)$ .

Worked Example

Example 1: Find the remainder when

f(x) = x^2 - 4x + 3

is divided by

x - 1

f(1) = (1)^2 - 4(1) + 3

= 1 - 4 + 3

= 0

The remainder is 0.

Example 2: Find the remainder when

f(x) = 2x^3 + 3x^2 - x + 1

is divided by

x + 2

.
Use

f(-2)

because

x + 2 = x - (-2)

f(-2) = 2(-2)^3 + 3(-2)^2 - (-2) + 1

= 2(-8) + 3(4) + 2 + 1

= -16 + 12 + 2 + 1 = -1

The remainder is -1.

Example 3: If

f(x) = x^3 - 2x^2 + x - 5

, find the remainder when divided by

x - 3

f(3) = (3)^3 - 2(3)^2 + 3 - 5

= 27 - 18 + 3 - 5 = 7

The remainder is 7.

Example 4: Determine if

x + 1

is a factor of

x^3 + 3x^2 + 3x + 1

.
Use

f(-1)

f(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1

= -1 + 3 - 3 + 1 = 0

Since the remainder is 0,

x + 1

is a factor.

Example 5: A polynomial

f(x) = 5x^4 - 3x^2 + x - 2

is divided by

x - 1

. Find the remainder.

f(1) = 5(1)^4 - 3(1)^2 + 1 - 2

= 5 - 3 + 1 - 2 = 1

The remainder is 1.

Exercises

$f(x) = x^2 - 2x + 1$ , find the remainder when divided by $x - 1$ .
$f(x) = x^3 + 2x^2 + x + 1$ , find the remainder when divided by $x + 1$ .
$f(x) = 3x^3 - x^2 + 4x - 7$ , find the remainder when divided by $x - 2$ .
[WAEC] $f(x) = x^3 - 6x + 7$ ; find the remainder when divided by $x - 1$ . (Past Question)
$f(x) = x^4 - 16$ ; what is the remainder when divided by $x + 2$ ?
[NABTEC] $f(x) = x^3 + 4x^2 + 5x + 2$ ; find the remainder when divided by $x - 3$ . (Past Question)
$f(x) = 2x^3 - x^2 + 3x - 1$ , find the remainder when divided by $x - 0.5$ .
[WAEC] $f(x) = x^3 - 9x + 8$ ; find the remainder when divided by $x + 2$ . (Past Question)
$f(x) = x^2 - 5x + 6$ ; determine if $x - 3$ is a factor.
[JAMB] $f(x) = 4x^3 + x - 6$ ; find the remainder when divided by $x - 2$ . (Past Question)

Conclusion / Recap

The Remainder Theorem is a fast way to evaluate the remainder of a polynomial division. If $f(a) = 0$ , then $x - a$ is a factor. In the next lesson, we’ll build on this by exploring the Factor Theorem.