| Supplementary Chapters |
| Content |
|---|
| Basic Counting |
| Polynomial Division |
| Partial Fractions |
| Summation Sign |
| Complex Numbers |
| Differentiation |
| Problems & Projects |
| Problem Set |
| Solutions |
| Exercise Solutions |
| Problem Set Solutions |
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 .
