What if you had a polynomial like
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CK12 Foundation: 0912S Factoring Polynomials Completely
The WTAMU Virtual Math Lab has a detailed page on factoring polynomials here: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm. This page contains many videos showing example problems being solved.
Guidance
We say that a polynomial is factored completely when we can’t factor it any more. Here are some suggestions that you should follow to make sure that you factor completely:
 Factor all common monomials first.
 Identify special products such as difference of squares or the square of a binomial. Factor according to their formulas.
 If there are no special products, factor using the methods we learned in the previous sections.
 Look at each factor and see if any of these can be factored further.
Example A
Factor the following polynomials completely.
a)
b)
c)
Solution
a) Factor out the common monomial. In this case 6 can be divided from each term:
There are no special products. We factor
The two numbers that multiply to 6 and add to 5 are 2 and 3, so:
If we look at each factor we see that we can factor no more.
The answer is
b) Factor out common monomials:
We recognize
If we look at each factor we see that we can factor no more.
The answer is
c) Factor out common monomials:
We recognize
If we look at each factor we see that we can factor no more.
The answer is
Example B
Factor the following polynomials completely:
a)
b)
Solution
a) Factor out the common monomial. In this case, factor out 2 rather than 2. (It’s always easier to factor out the negative number so that the highest degree term is positive.)
We recognize expression in parenthesis as a difference of squares. We factor and get:
If we look at each factor we see that the first parenthesis is a difference of squares. We factor and get:
If we look at each factor now we see that we can factor no more.
The answer is
b) Factor out the common monomial:
We recognize
We look at each term and recognize that the term in parentheses is a difference of squares.
We factor it and get
If we look at each factor now we see that we can factor no more.
The final answer is
Factor out a Common Binomial
The first step in the factoring process is often factoring out the common monomials from a polynomial. But sometimes polynomials have common terms that are binomials. For example, consider the following expression:
Since the term
This expression is now completely factored.
Let’s look at some more examples.
Example C
Factor out the common binomials.
a)
b)
Solution
a)
When we factor out the common binomial we get
b)
When we factor out the common binomial we get
Watch this video for help with the Examples above.
CK12 Foundation: Factoring Polynomials Completely
Vocabulary
 We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more.
Guided Practice
Factor completely:
Solution:
First, notice that each term has
Next, factor the trinomial in the parenthesis. Since
Rewrite the trinomial using
The final factored answer is:
Explore More
Factor completely.

2x2+16x+30 
5x2−70x+245 
−x3+17x2−70x 
2x4−512 
25x4−20x3+4x2 
12x3+12x2+3x 
12c2−75 
6x2−600 
−5t2−20t−20 
6x2+18x−24 
−n2+10n−21 
2a2−14a−16