First of all, we need to incorporate some notions about a much more fundamental concept: factoring.
We can factor numbers,
or even expressions involving variables (polynomials),
Factoring is the process of splitting an expression into a product of simpler expressions. It's a technique we'll be using a lot when working with polynomials.
There are some cases where dividing polynomials may come as an easy task to do, for instance:
Another trickier example making use of factors:
One more time,
1. Try dividing by .
2. Now, can you factor ?
What about a non-divisible polynomials? Like these ones:
Sometimes, we'll have to deal with complex divisions, involving large or non-divisible polynomials. In these cases, we can use the long division method to obtain a quotient, and a remainder:
In this case:
|Long division method
||We first consider the highest-degree terms from both the dividend and divisor, the result is the first term of our quotient.
||Then we multiply this by our divisor.
||And subtract the result from our dividend.
||Now once again with the highest-degree terms of the remaining polynomial, and we got the second term of our quotient.
||We are left with a constant term - our remainder:
3. Find some such that is divisible by .