Long division of whole numbers was a major topic at school before the days of calculators and long division in algebra is a very similar process. You may have covered this earlier. The two processes are placed side by side below for comparison. See if you can follow the working.

To calculate 3724 ÷ 15 |
To calculate x^{3} + 3x^{2} + 6x + 5 ÷ (x + 1) |
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This means: |
This means: |
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Terminology |
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dividend |
the expression being divided. | 3724 | x^{3} + 3x^{2} + 6x + 5 |

divisor |
the expression being divided in. | 15 | x + 1 |

quotient |
the answer | 248 | x^{2} + 2x + 4 |

remainder |
the left overs | 4 | 1 |

### The Remainder Theorem

In the second example above, the remainder is 1.

This remainder can be worked out much quicker without the long division using the** Remainder Theorem.**

The Remainder Theorem states:

If a polynomial f(x) is divided by (x − a) then the remainder is f(a). |

For the example above: When f(x) = x^{3} + 3x^{2} + 6x + 5

is divided by (x + 1) the remainder is f(-1)

Remainder is f(-1) = (-1)^{3} + 3(-1)^{2} + 6(-1) + 5 =** 1**

A much quicker way to find the remainder!