# 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:

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

When \(p(x)\) is divided by \((x-a)\): remainder\(=p(a)\)

OR

When \(p(x)\) is divided by \((ax+b)\): remainder\(=p\left(-\frac{b}{a}\right)\)

** 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 \((x^6+4x^5+9x^3-4x^2+10) \div (x+1)\).

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

## Related to This Article

### More math articles

- Top 10 AFOQT Math Practice Questions
- The Best Math Apps to Make Learning Easier
- 4th Grade SBAC Math Practice Test Questions
- Free GED Math Worksheets [Updated for 2024]
- Top 10 Tips to Create an ASVAB Math Study Plan
- The Ultimate 6th Grade IAR Math Course (+FREE Worksheets)
- Exponential and logarithmic integrals Simplified
- 8th Grade PSSA Math Practice Test Questions
- Intelligent Math Puzzle – Challenge 88
- Top 10 5th Grade OST Math Practice Questions

## What people say about "The Remainder Theorem - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.