Math 101 Intermediate Algebra

Factoring a Monomial from a Polynomial and Factoring by Grouping
Chapter 6, Section 1

Finding the Greatest Common Factor (GCF) of Terms

Given a term c, if a · b = c then a and b are factors of c.

Example:   What are some factors of the term 6x3?
Factors Because...
6 and x3 6 · x3 = 6x3
2 and 3x3 2 · 3x3 = 6x3
-2 and -3x3 -2 · -3x3 = 6x3
2x and 3x2 2x · 3x2 = 6x3
-2x and -3x2 -2x · -3x2 = 6x3
6x and x2 6x · x2 = 6x3
-6x and -x2 -6x · -x2 = 6x3
The above factors are not the only factors:
there are more factors of 6x3.

The greatest common factor (GCF) of two or more expressions is the greatest factor that divides into (without remainder) each expression.

The GCF of a bunch of terms contains the lowest power of the variable common to all the terms.

Steps to Factoring a Monomial from a Polynomial

  1. Determine the GCF of all terms in the polynomial.
  2. Write each term as the product of the GCF and another factor.
  3. Use the distributive property to factor out the GCF.

The first step in any factoring problem is to factor out the GCF.

Factoring a 4 Term Polynomial by Grouping

  1. Arrange the 4 terms into 2 groups of 2 terms each so that each group of 2 terms has a GCF.
  2. Factor the GCF from each group of 2 terms.
  3. If the two, new terms formed by step 2 have a GCF, then factor it out.


When you multiply out the result of factoring,
you must get the original expression you're trying to factor.


The book contains lots of good examples for you to look at.