• # Revision:Algebraic Long Division

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Algebraic long division is very similar to traditional long division (which you may have come across earlier in your education).

## Contents

### Example

We want to simplify The procedure is as follows:

Step 1:

Set up the problem like a normal long division problem as above.

Step 2:

Divide the first term of the numerator by the first term of the denominator. In this case, .

Step 3:

Multiply this result by the whole of the denominator. Write this underneath. So here: .

Step 4:

Subtract: . Bring the next term "down".

Step 5:

Divide the first term of the new expression by the first term of the denominator. (Note that we have, at this stage, deduced that , and are now looking to divide out the smaller fraction; then step 5 is equivalent to step 2.)

Step 6:

Multiply as in step 3.

Step 7:

Continue. Note that, when we have finished, we end up with a 0 - that is, no remainder. This won't be the case for all polynomial division, we were just lucky!

NB: If the numerator has a term in x missing, add the missing term by placing a zero in front of it. For example, if you are dividing by something, rewrite it as .

For algebraic long division practice makes perfect- the best way to learn how to do them properly is to do loads of examples until you get them right every time!

## The Remainder Theorem

When dividing one algebraic expression by another, more often than not there will be a remainder. It is often useful to know what this remainder is and it can be calculated without going through the process of dividing as above. The rule is:

• If a polynomial is divided by , the remainder is

### Example

In the above example, was divided by .

Let . The remainder is therefore , as we saw when we divided the whole thing out. However, dividing by would have given us a remainder of .

## The Factor Theorem

This states:

• If is a factor of the polynomial , then . Conversely, if and f is a polynomial function, then is a factor of .

### Example

In the above worked example, . This means that is a factor of the equation.