Factoring Using the Greatest Common Factor (GCF)

Between two or more numbers, terms or polynomials, the greatest common factor (GCF) is the largest value that evenly divides them all. For example, the GCF of the integers 6 and 10 is 2 because 2 is the largest integer that evenly divides both. The process of finding the GCF of two or more integers is helped by using prime factorization. Prime factorization is just the process of breaking down an integer into a number of factors until they are all prime numbers. They have to be prime numbers because prime numbers can’t be broken down into any more factors. Here’s an example of finding the GCF of two larger integers using prime factorization:

Find the GCF of 24 and 36.
24 = 2 * 12 = 2 * 2 * 6 = 2 * 2 * 2 * 3
36 = 6 * 6 = 2 * 3 * 2 * 3 = 2 * 2 * 3 * 3
The common factors are 2, 2, and 3, so the GCF is 2 * 2 * 3 = 12.

We can expand the idea of finding the GCF of two or more things by looking at polynomial terms. As you learned previously, terms have a coefficient portion and a variable portion. When finding the GCF mentally, these portions are typically done separately. Here’s an example of doing it the long way:

$Find\;the\;GCF\;of\;4x^2y^3\;and\;6x^3y^4.$
$4x^2y^3 = 2 * 2 * x * x * y * y * y$
$6x^3y^4 = 2 * 3 * x * x * x * y * y * y * y$
$The\;common\;factors\;are\;2,\;x,\;x,\;y,\;y,\;and\;y.$
$The\;GCF\;is\;then\;2 * x * x * y * y * y = 2x^2y^3.$

We can use the GCF of two polynomial terms to factor a polynomial made up by adding them together. Remember that factoring just means finding two or more values that multiply together to give you a set product. Consider this example:

$Factor\;4x^2y^3 + 6x^3y^4$
$The\;GCF\;of\;4x^2y^3\;and\;6x^3y^4\;is\;2x^2y^3.$
$4x^2y^3 + 6x^3y^4$
$= 2x^2y^3(\frac{4x^2y^3}{2x^2y^3} + \frac{6x^3y^4}{2x^2y^3})$
$= 2x^2y^3(2y + 3xy)$

The idea is that we want to use the distributive property to “pull out” the GCF from each term in the original polynomial that we’re trying to factor.