Remainder Theorem

Related Pages
Factor Theorem
Solving Cubic Equations
Remainder And Factor Theorems
More Algebra Lessons

Remainder Theorem And Factor Theorem

The following table gives the Remainder Theorem and Factor Theorem. Scroll down the page for more examples and solutions on how to use the Remainder Theorem and Factor Theorem.

What is the Remainder Theorem?
Recall that for long division for integers, the dividing process stops when the remainder is less than the divisor.

dividend = divisor × quotient + remainder

The process is similar for division of polynomials. The dividing stops when the remainder is less that the degree of the divisor.

The Remainder Theorem states that:

If a polynomial f(x) is divided by a linear divisor (x – a), the remainder is f(a)

Hence, when the divisor is linear, the remainder can be found by using the Remainder Theorem.

How to use the Remainder Theorem?

Example:
Find the remainder when 4x³ – 5x + 1 is divided by
a) x – 2
b) x + 3
c) 2x – 1

Solution:
Let f(x) = 4x³– 5x + 1
a) When f(x) is divided by x – 2, remainder,
R = f(2) = 4(2)³– 5(2) + 1 = 23

b) When f(x) is divided by x + 3, remainder,
R = f(–3) = 4(–3)³– 5(–3) + 1 = –92

c) When f(x) is divided by 2x – 1, remainder,

Example:
The expression 4x² – px + 7 leaves a remainder of –2 when divided by x – 3. Find the value of p.

Solution:
Let f(x) = 4x²– px + 7

By the Remainder Theorem,
f(3) = –2
4(3)²– 3p + 7 = –2
p = 15

How to use the Remainder Theorem to find the remainder?

Examples:
Use the Remainder Theorem to find the remainder

1. (3x³ - 2x² + x - 6) ÷ (x - 4)
   
2. (-4x³ + 8x² + 12x + 16) ÷ (x + 2)
   
3. (x⁶ + 4x⁵ + 9x³ - 4x² + 10) ÷ (x + 1)

• Show Video Lesson

How to find the remainder in a polynomial division?

Examples:
Find the remainder using the Remainder Theorem
(4x⁴ + 2x² + 1) ÷ (x - 3)

• Show Video Lesson

Finding The Remainders Of Polynomial Divisions

How to use the remainder and factor theorem in finding the remainders of polynomial divisions and also the factors of polynomial divisions?

The Remainder Theorem states that if a polynomial f(x) is divided by (x - k) then the remainder r = f(k). It can assist in factoring more complex polynomial expressions.

The Factor Theorem states that a polynomial f(x) has a factor (x - k) if and only f(k) = 0. It is a special case of the Remainder Theorem where the remainder = 0.

• Show Video Lesson

How to use the Remainder Theorem to test the factor of a polynomial?

Example:
Is x - 3 a factor of 2⁴ - 11x³ + 15x² + 4x - 12?

• Show Video Lesson

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.