High School Mathematics Extensions/Supplementary/Polynomial Division
Supplementary Chapters |
Content |
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Basic Counting |
Polynomial Division |
Partial Fractions |
Summation Sign |
Complex Numbers |
Differentiation |
Problems & Projects |
Problem Set |
Solutions |
Exercise Solutions |
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 | |||
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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 .