In this lesson we are going to learn how to factor an algebraic expression using the greatest common factor.
First let's review some vocabulary.
An algebraic expression is a collection of terms, all connected by either a "+" or a "-" sign.
For example:
3x + 9y - 12
15x² - 25x + 30
6ac - 18ad + 12ar
are all algebraic expressions.
The greatest common factor is the largest monomial term which will divide evenly into all of the terms in the expression
For example:
The gcf for 6x and 8x is 2x
because 2x is the largest monomial term
which will divide evenly into both 6x and 8x.
Another example:
The gcf for 35x³, 40x⁵, and 25x⁴ is 5x³
Let's look at one more example:
The gcf for 12x²y³z, 36x⁶y⁴c, and 24x⁶y⁵d is 12x²y³
Now let's look at what one of these
problems would look like:
Factor: 6a - 4b + 8c
The gcf is "2"
(none of the variables are in all of the terms, therefore they are not "common")
So the solution is written as:
2(3a - 2b + 4c)
Do you see that the solution is written
as the product of the gcf and the
original expression after you "factored out" the gcf.

Let's do another example:
Factor: 28x⁴y³ - 42x³y⁵
The gcf is 14x³y³
(14 is the gcf for 28 and 42, x³ and y³ are common in both of the terms)
So the solution is written:
14x³y³(2x - 3y²)

And let's do one more....
Factor: 5a³cd + 15a²c⁴e - 10a⁶c²f
The gcf is: 5a²c
(d, e, and f are not common factors because they are not in all of the terms!)
So the solution is:
5a²c(ad + 3c³e - 2a⁴cf)

For Practice
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