What if you had a polynomial like with multiple factors? How could you factor it completely? After completing this Concept, you'll be able to factor out common monomials and binomials from polynomials like this one.
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CK-12 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 as a product of two binomials:
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 as a difference of squares. We factor it as .
If we look at each factor we see that we can factor no more.
The answer is .
c) Factor out common monomials:
We recognize as a perfect square and factor it as .
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 as a perfect square and we factor it as .
We look at each term and recognize that the term in parentheses is a difference of squares.
We factor it and get , which we can rewrite as .
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 appears in both terms of the polynomial, we can factor it out. We write that term in front of a set of parentheses containing the terms that are left over:
This expression is now completely factored.
Let’s look at some more examples.
Example C
Factor out the common binomials.
a)
b)
Solution
a) has a common binomial of .
When we factor out the common binomial we get .
b) has a common binomial of .
When we factor out the common binomial we get .
Watch this video for help with the Examples above.
CK-12 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 as a factor. Start by factoring out :
Next, factor the trinomial in the parenthesis. Since find : . Find the factors of 12 that add up to -7. Since 12 is positive and -7 is negative, the two factors should be negative:
Rewrite the trinomial using , and then factor by grouping:
The final factored answer is:
Practice
Factor completely.