The opposite of multiplying polynomials is *
factoring*. Why would you want to factor a polynomial?

Let be a polynomial. is equivalent to dividing .

Recall that when , you say that c is a * root* of . The result above
means that factoring is related to root-finding, and vice versa.

* Example.* If ,

It follows that divides ; in fact,

Besides root-finding, you may also want to factor an expression in order to simplify something.

* Example.* Provided that , I can simplify by cancellation:

I got the last equality by cancelling the 's, which is valid provided that . I "found" the on the top by factoring.

I'll look at various methods for factoring polynomials: Removing a common factor, factoring quadratics by trial and error, using special forms for quadratics, using special forms for cubics, and factoring by grouping.

* 1. Removing a common factor.*

In some cases, the terms of a sum have some factors in common. The common factors may be factored out of all the terms:

This is the opposite of the distributive law.

* Example.*

* 2. Factoring quadratics.*

Now I'll discuss the important case of * factoring
quadratics*. The discussion will be incomplete until later, when
I discuss the * general quadratic formula*.

* Example.* How do you factor ?

To do this, think of two numbers which multiply to 12 and add to give 7. 3 and 4 work: and . So

How do you factor ?

Think of two numbers which multiply to 6 and add to give -5. I must have two negative numbers to make this work; -2 and -3 fit the bill. So

How do you factor ?

Think of two numbers which multiply to -6 and add to give -5. I must have two negative numbers to make this work; -6 and 1 fit the bill. So

How do you factor ?

Think of two numbers which multiply to 16 and add to give -8. I must have two negative numbers to make this work; -4 and -4 fit the bill. So

I have factoring formulas which correspond to my special forms for multiplication.

1.

2.

3.

* Example.* This is the form with and :

This is the form with and :

This is the form with and :

In the next example, I'll use the form with and :

Here's the form with and :

This example uses the form with and :

And this example uses the form with and :

Note, however, that does *not* factor
(unless you use complex numbers).

* Example.* How do you factor ?

I expect a factorization that looks like this:

What are the missing terms? I write down all possible ways of factoring -5, assuming that this will come out in terms of integers:

I find that

You can check this by multiplying out the right side.

* Example.* Factor .

I need numbers a and b whose product is 12, and such that

If you try various combinations, you'll find that

* Examples.* Factoring quadratics is something
you should be able to do fluently. Here are some more for practice.

To factor , I need two numbers which add to 9 and multiply to 14. The pairs of numbers multiplying to 14 are and . , so I use 2 and 7:

To factor , I need two numbers which add to -9 and multiply to 18. The pairs of numbers which multiply to 18 are , , , , , and , , so I use -3 and -6:

To factor , you can look for two numbers which add to -14 and multiply to 49. -7 and -7 work.

I would do it differently: The perfect square 49 makes me think that maybe this is a standard form. , and , which is the middle coefficient --- and that led me to . I used -7 because the middle coefficient was -14. So either way,

To factor , I need two numbers which add to -2 and multiply to -24. The pairs of numbers which multiply to -24 are , , , , , , , and . , so

Don't forget that you can always check your factoring by multiplication!

To factor , I notice that both the 4 in and the 1 are perfect squares. and ; is this a standard form? Well, , so

To factor , I notice that breaks down as and 3 could be either or . To get the -5 in the middle, I must have -1 and -3. So there are two possibilities:

In fact,

is more complicated. The could be or . The -3 is either or . Here are the possibilities:

So

* Example.* Factor .

First,

*Fact:* A sum of two squares does not factor. Thus, can't be factored. The
factorization is .

* Example.* (* More than one
variable*) Factor .

I take out a common factor, then factor the remaining quadratic term by trial:

Notice that is like , except with the additional b's.

* 3. Cubic formulas.*

1.

2.

* Example.*

* 4. Factoring by grouping.*

In some cases, you can factor an expression *by factoring pieces
of the expression separately, then looking for common factors in the
pieces*. This is easier to show than to explain, so here are some
examples.

* Example.* Factor .

I don't have a rule for factoring a cubic of this form. I'll break the polynomial up into two pieces:

Now I'll take a common factor out of each piece, then look for a common factor of the whole expression.

* Example.* Factor .

* Example.* Factor .

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Copyright 2013 by Bruce Ikenaga