# The Remainder Theorem

The remainder theorem is a formula used to find the remainder when a polynomial is divided by a linear polynomial. In this step-by-step guide, you learn more about the remainder theorem.

When a certain number of things are divided into groups with an equal number of things in each group, the number of things left is known as the remainder. This is what “remains” after division.

## Step by step guide to the remainder theorem

The **remainder theorem** is expressed as follows:

When a polynomial \(a(x)\) is divided by a linear polynomial \(b(x)\) whose zero is \(x = k\), the remainder is obtained by \(r = a (k)\). The remaining theorem enables us to compute the remainder of dividing any polynomial by a linear polynomial without performing the steps of the division algorithm.

### Remainder theorem formula

The general formula of the remainder theorem is given as follows:

\(\color{blue}{p(x)=(x-c).q(x)+r(x)}\)

**Important notes**:

- When a polynomial \(a(x)\) is divided by a linear polynomial \(b(x\)) whose zero is \(x = k\), the remainder is given by \(r = a(k)\)
- The remainder theorem formula is: \(p(x)=(x-c).q(x)+r(x)\)
- The basic formula to check the division is:
**Dividend**\(=\) (**Divisor**\(×\)**Quotient**) \(+\)**Remainder**

### The Remainder Theorem – Example 1:

Find the remainder when \(p(x)=3x^5-x^4+x^3-4x^2+2\) is divided by \(q(x):x-1\).

Replace the zero of \(q(x)\) into the polynomial \(p(x)\) to find the remainder \(r\):

\(x-1=0 → x=1\)

\(p(1)=3(1)^5-(1)^4+(1)^3-4(1)^2+2\)

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

\(=1\)

Therefore, the remainder is \(1\).

### The Remainder Theorem – Example 2:

Find the remainder when \(p(x)=x^3-x^2+x-1\) is divided by \(q(x):x+1\).

Replace the zero of \(q(x)\) into the polynomial \(p(x)\) to find the remainder \(r\):

\(x+1=0 → x=-1\)

\(p(-1)=(-1)^3-(1)^2+(-1)-1\)

\(=-1-1-1-1\)

\(=-4\)

Therefore, the remainder is \(-4\).

## Exercises for the Remainder Theorem

- Find the remainder after \(2x^2-5x-1\) is divided by \(x-5\).
- Use the remainder theorem to evaluate \(f(x)=2x^5+4x^4-3x^3+8x^2+7\) at \(x=2\).
- Find the remainder when \(4x^3-5x+1\) is divided by \(2x-1\).
- Use the remainder theorem to find the remainder \((x^6+4x^5+9x^3-4x^2+10) \div (x+1)\).

- \(\color{blue}{24}\)
- \(\color{blue}{143}\)
- \(\color{blue}{-1}\)
- \(\color{blue}{-6}\)

### More math articles

- How to Graph Rational Functions?
- PSAT 10 Math Practice Test Questions
- How to Prepare for the OAR Math Test?
- 10 Most Common 5th Grade Georgia Milestones Assessment System Math Questions
- Full-Length 6th Grade GMAS Math Practice Test-Answers and Explanations
- The Best Calculator for the SAT or ACT
- The Ultimate SSAT Middle Level Math Formula Cheat Sheet
- 5 Signs to Understand You’re Good at Math
- How to Interpret Pie Graphs? (+FREE Worksheet!)
- The Ultimate ALEKS Math Course (+FREE Worksheets & Tests)

## What people say about "The Remainder Theorem"?

No one replied yet.