In these lessons, we will look at the Remainder Theorem and how it relates to the Factor Theorem. We will also show how to solve polynomial problems using the Remainder Theorem.

**Related Pages**

Factor Theorem

Solving Cubic Equations

Remainder And Factor Theorems

More Algebra Lessons

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 4x^{3} – 5x + 1 is divided by

a) x – 2

b) x + 3

c) 2x – 1

**Solution:**

Let f(x) = 4x^{3}– 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 4x^{2} – px + 7 leaves a remainder of –2 when divided by x – 3.
Find the value of p.

**Solution:**

Let f(x) = 4x^{2}– 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

- (3x
^{3}- 2x^{2}+ x - 6) ÷ (x - 4) - (-4x
^{3}+ 8x^{2}+ 12x + 16) ÷ (x + 2) - (x
^{6}+ 4x^{5}+ 9x^{3}- 4x^{2}+ 10) ÷ (x + 1)

**How to find the remainder in a polynomial division?**

**Examples:**

Find the remainder using the Remainder Theorem

(4x^{4} + 2x^{2} + 1) ÷ (x - 3)

**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.

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

**Example:**

Is x - 3 a factor of 2^{4} - 11x^{3} + 15x^{2} + 4x - 12?

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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