Algebraic long division is very similar to traditional long division (which you may have come across earlier in your education).
Set up the problem like a normal long division problem as above.
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.)
Multiply as in step 3.
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!
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:
The Factor Theorem
- If is a factor of the polynomial , then . Conversely, if and f is a polynomial function, then is a factor of .