High School Mathematics Extensions/Supplementary/Polynomial Division

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Supplementary Chapters
Content
75% developed Basic Counting
50% developed Polynomial Division
100% developed Partial Fractions
75% developed Summation Sign
75% developed Complex Numbers
75% developed Differentiation
Problems & Projects
0% developed Problem Set
Solutions
0% developed Exercise Solutions
0% developed Problem Set Solutions

Introduction[edit]

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[edit]

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[edit]

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  .