Of all the topics covered in this chapter factoring polynomials is probably the most important topic. There are many sections in later chapters where the first step will be to factor a polynomial. So, if you can’t factor the polynomial then you won’t be able to even start the problem let alone finish it.
Let’s start out by talking a little bit about just what factoring is. Factoring is the process by which we go about determining what we multiplied to get the given quantity. We do this all the time with numbers. For instance, here are a variety of ways to factor 12.
There are many more possible ways to factor 12, but these are representative of many of them.
A common method of factoring numbers is to completely factor the number into positive prime factors. A prime number is a number whose only positive factors are 1 and itself. For example 2, 3, 5, and 7 are all examples of prime numbers. Examples of numbers that aren’t prime are 4, 6, and 12 to pick a few.
If we completely factor a number into positive prime factors there will only be one way of doing it. That is the reason for factoring things in this way. For our example above with 12 the complete factorization is,
Factoring polynomials is done in pretty much the same manner. We determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This continues until we simply can’t factor anymore. When we can’t do any more factoring we will say that the polynomial is completely factored.
Here are a couple of examples.
This is completely factored since neither of the two factors on the right can be further factored.
is not completely factored because the second factor can be further factored. Note that the first factor is completely factored however. Here is the complete factorization of this polynomial.
The purpose of this section is to familiarize ourselves with many of the techniques for factoring polynomials.
Greatest Common Factor
The first method for factoring polynomials will be factoring out the greatest common factor. When factoring in general this will also be the first thing that we should try as it will often simplify the problem.
To use this method all that we do is look at all the terms and determine if there is a factor that is in common to all the terms. If there is, we will factor it out of the polynomial. Also note that in this case we are really only using the distributive law in reverse. Remember that the distributive law states that
In factoring out the greatest common factor we do this in reverse. We notice that each term has an a in it and so we “factor” it out using the distributive law in reverse as follows,
|Example 1 Factor out the greatest common factor from each of the following polynomials.
First we will notice that we can factor a 2 out of every term. Also note that we can factor an x2 out of every term. Here then is the factoring for this problem.
Note that we can always check our factoring by multiplying the terms back out to make sure we get the original polynomial.