Factoring by Grouping: Help with Four Term Polynomials

Updated on June 18, 2013

Dr. Jerry Allison is founder of Kairos Advising and Consulting and has worked with businesses and teaching students business for 30+ years.

One of the most useful topics in algebra is factoring by grouping. To many students it does not seem like an important topic, but to some it is extremely important. Another article describes how factoring by grouping is used to factor trinomials. This article describes the process and provides some examples.

The Factoring by Grouping Process

Below are the steps to accomplish factoring by grouping. It is assumed that there is no greatest common factor (GCF).

1. Put parentheses around the first two terms and around the last two terms. When putting in these parentheses, one must be careful to make sure negative signs go inside the parentheses. So it may be necessary to rewrite a subtraction as an addition.
2. Factor the GCF from the first set of parentheses and from the second set of parenthesis. These will be different GCFs. However, if the original four term polynomial is factorable, then what is left inside both sets of parentheses will be exactly the same.
3. Factor out the common binomial.

In the next section are a few examples of the process.

Examples of Using Factoring by Grouping

Example 1:

Factor x3 +3x2 + 5x + 15.

Step one says put parentheses around the first two terms and around the last two terms. Since there are no negative signs, that is not a concern.

(x3 +3x2) + (5x + 15)

Step two says factor the GCF from each set of parentheses. In the first set of parentheses, x2 is the GCF; in the second, the GCF is 5.

x2 (x + 3) + 5 (x + 3)

Now what is in the parentheses should be exactly the same. It they are not the same, then the polynomial is not factorable or there has been a mistake made.

Step three now says factor out the common binomial.

(x + 3) ( x2 + 5)

This is the answer. This answer can be checked by multiplying the binomials back out to get the original problem.

Example 2:

Factor 2x2 + 2ax - xb - ab.

Step one says put parentheses around the first two terms and around the last two terms. However, the negative sign in the middle creates a difficulty. So it is best to rewrite the subtraction in the middle as an addition problem.

2x2 + 2ax + - xb - ab

Now put the parentheses around the first two terms and around the last two.

(2x2 + 2ax) + (-xb - ab)

Step two states factor the GCF from each set of parentheses.

2x (x + a) - b (x + a)

Step three state factor out the common binomial.

(x + a) (2x - b)

Conclusion

Factoring by grouping is a useful technique for factoring four term polynomials. As stated in another article, its use can be extended to factoring trinomials. It is a process that is worth mastering.

Addition information may be found on Dr Allison's Website.

3. Math.com

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