High School Mathematics Extensions/Supplementary/Polynomial Division

Introduction

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.

Dividing polynomials

There are some cases where dividing polynomials may come as an easy task to do, for instance:

Distributing,

Finally,

Another trickier example making use of factors:

Reordering,

Factoring,

One more time,

Yielding,

1. Try dividing  by  .

2. Now, can you factor  ?

Long division

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
1	We first consider the highest-degree terms from both the dividend and divisor, the result is the first term of our quotient.
2	Then we multiply this by our divisor.
3	And subtract the result from our dividend.
4	Now once again with the highest-degree terms of the remaining polynomial, and we got the second term of our quotient.
5	Multiplying...
6	Subtracting...
7	We are left with a constant term - our remainder:

So finally:

3. Find some  such that  is divisible by  .