## Long Division and the Remainder Theorem

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 x3 + 3x2 + 6x + 5 ÷ (x + 1)  This means: This means: Terminology dividend the expression being divided. 3724 x3 + 3x2 + 6x + 5 divisor the expression being divided in. 15 x + 1 quotient the answer 248 x2 + 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) = x3 + 3x2 + 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!