Factoring Polynomials by Finding the Greatest Common
Factor (GCF)
To factor means to write a quantity or an expression as a product.
To factor a whole number, such as 20, write it as a product of whole
numbers. For example:
20 = 2 · 10
20 = 4 · 5
20 = 2 · 2 ·
5
To factor a polynomial, such as x2 + 3x, means to write it as a product of
polynomials. For example:
x2 + 3x = x(x + 3)
We say x(x + 3) is the factorization of x2 + 3x.
To factor a polynomial, first find its greatest common factor (GCF). The
GCF contains all of the factors that are common to all of the terms
Finding the Greatest Common Factor (GCF) of a Set of Monomials
Procedure —
To Find the Greatest Common Factor (GCF) of a Set of Monomials
Step 1 Factor each monomial.
Step 2 List each common factor the least number of times it
appears in any factorization.
Step 3 Multiply the factors in the list.
If the monomials have no common factors, other than 1, then the
GCF is 1.
Example
Find the GCF of 36x3y2, 12x2y4, and 24x2y.
Solution
| Step 1 Factor each monomial.
Step 2 List each common factor the
least number of times it appears
in any factorization.
|
36x3y2 = 2 · 2 · 3 · 3 · x · x · x · y ·
y
12x2y4 = 2 ·
2 · 3 ·
x · x ·
y · y ·
y · y
24x2y = 2 · 2 ·
2 · 3 ·
x · x ·
y
2, 2, 3, x, x, y |
| Step 3 Multiply the factors in the list. |
2 · 2 ·
3 · x ·
x · y = 12x2y |
Thus, the GCF is 12x2y.
Note:
As a check, note that the GCF, 12x2y, is a
factor of each of the given monomial
terms.
36x2y2 = 12x2y · 3xy
12x2y4 = 12x2y · y3
24x2y = 12x2y · 2
Also, note that the GCF of the remaining
factors, 3xy, y3, and 2 is 1.
|