Factoring:
We can multiply polynomials such as 
by using the FOIL process to obtain a product 
. Values we multiply together are called factors, so 
and 
are factors. The result obtained from multiplying the factors is called a product, so the polynomial,
, is the resulting product. Now we reverse the process to write a polynomial as a product of factors. This process of taking a polynomial and writing it as a product of factors is called factoring. We generally factor using integers. (Integers are the whole numbers and their opposites.) If the polynomial cannot be factored, we say that the polynomial is prime.
If the instructions say to factor, this means that we are to find the complete factored form. This means that we are to factor the expression as completely as possible using integer values.
One of the first things we look for in factoring is whether we can factor out a common factor. Notice the terms 
and 
in the expression 
share a common factor of 3 or 
. When you factor out the greatest common factor (abbreviated as GCF), you can factor out a negative sign or leave the negative sign inside the parentheses.
Example: Factor out the GCF: 
We can factor out a 3 from each term of the given expression and in doing so, we divide each term by what we factor out, in this case 3.
Notice that our factored result could also be written as 
since 
can be written as 
. It is helpful to see that the factor 
is the same as 
. They are just written in different forms. It is similar to recognizing that 
is equivalent to 
.
We could also factor out a 
from each term of the given expression.
Any of these answers 
or 
or 
are acceptable.
In factoring out the greatest common factor, there are occasions when factoring out a negative sign may be preferable so that the polynomial can be factored completely.
Example: Factor out the GCF: 
To factor the greatest common factor from , first notice that 
and 
are not equal and we cannot factor. But we could factor a 
out from to obtain 
which can be written as 
so that 
may be factored out:
Notice in the line above that is shared by two terms. We may now factor out the .
Take the time to memorize the special polynomial forms involving difference of two squares, sum of two cubes, and difference of two cubes. These forms occur frequently throughout the remainder of this course and you will save yourself time in factoring by recognizing these forms and knowing the formulas.
Difference of squares: 
Difference of cubes: 
Sum of cubes: 
If you are asked to factor an expression that involves four terms, the most appropriate method is to use grouping. Try grouping together the first pair of terms and the second pair of terms with addition separating the first pair from the second pair. Factor any GCF.
Example: Factor by grouping: 
We group the first pair of terms and the second pair of terms.
In the grouping of the first pair of terms, 
, is a common factor. In the second grouping, 
, is a common factor. We factor these out.
Now notice that 
is a common factor, and we factor this out.
Most teachers use a trial and error method to factor an expression involving three terms. There is an alternative method that does not use as much trial and error. To use this "new" method, the trinomial is written in a way that factoring by grouping can be done. If you struggle with the trial and error method, you may find this method more convenient.
These steps can be used to factor a trinomial of the form 
(where a, b, and c are integers and 
).
Step 1: Multiply a and c together, forming the product ac.
Step 2: Find two numbers that multiply to form ac, but when added form b.
Step 3: Rewrite the polynomial so its middle term bx is written as a sum of two terms whose coefficients are the two numbers found in step 2.
Step 4: Factor by grouping.
Example: Factor 
using the above steps.
The trinomial has the form where 
.
Step 1: Multiply ac gives 
.
Step 2: We want to find two numbers that multiply to 
and when added make 
. Look at pairs of numbers whose product is 
and then find their sum. (If you use this method, you do not have to make an entire list of products and sums as was done below for this problem. With some practice you can do this step mentally.)
Product Sum
Notice only one pair 3 and 
have a sum of . If there were no pair whose product is ac and whose sum is b, then the polynomial is not factorable with integer coefficients and is considered prime.
Step 3: Rewrite the polynomial so the middle term 
is written as a sum of 
and 
.
Step 4: Use the factor by grouping method.
You can check the factorization by multiplying 
to see if you obtain the polynomial .
Here is a listing of steps you can use to factor polynomials.
Step 1: Always factor out any greatest common factor.
Step 2: If the polynomial is a binomial, see if it is a difference of two squares or the difference of two cubes, or the sum of two cubes, and factor accordingly. (Notice there is no formula for factoring the sum of two squares, 
.)
Step 3: If the polynomial is a trinomial, use the trial and error method or the alternative method given for factoring.
Step 4: If the polynomial has more than three terms, try to factor by grouping.
Step 5: Always check to see if any factors you have found can be factored further.
Always check for a common factor first before using any other methods of factoring. In the long run, it makes the process of factoring much easier!
Example: Factor completely: 
If you factor 
into 
, the factoring process is harder and you would have to factor 8 from 
to obtain the correct answer. You may as well factor 8 first thing.