Factoring is the method of breaking down a number or expression into products of smaller parts. We call these smaller parts the factors of an expression. When the factors of an expression are all multiplied together they create the original expression (for example, the factors of 15 are 3 and 5 because 3*5=15). Factoring has many uses in simplifying expressions and solving equations. We’ll see the applications of factoring later, but first let’s learn some of the common factoring techniques…

Greatest Common Factor
The Greatest Common Factor is the largest number that is a common factor of two or more numbers. A common use of using the greatest common factor is in simplifying fractions. You can simplify a fraction by finding the greatest common factor between the numerator and denominator, and then dividing both by that number. For example, to reduce 3/6 you’ll need to find the GCF (greatest common factor) between the numerator and denominator (3) and then divide the numerator and denominator by that number. In this example, divide the numerator and denominator by 3 to get the simplified fraction: 1/2.

Factoring a set of expressions is similar to the method used for fractions. The goal is to find the GCF for a set of expression, then divide each term by this GCF to find the factors. Let’s see an example:

12x+18

First, we calculate the prime factors of the integers in the expression:

3*2*2x + 3*3*2

Now that both terms in the expression are broken down, we can look for common factors between them. As you can see, both terms have a factor of 3 and a factor of 2. To factor the expression, pull these factors out of each term…

3*2 (2x + 3)

and then multiply them together…

6(2x+3)

That’s it, we’ve factored the expression. Now let’s look at an example that is a little more complicated:

3x²y²+9x³y+6x³y²

First, break down every term into its simplest factors:

3*x*x*y*y + 3*3*x*x*x*y + 2*3*x*x*x*y*y

Now go through the terms to locate common factors:

(3*x*x*y)*y + (3*x*x*y)*3x + (3*x*x*y)*2xy

So we’ve found the greatest common factor to be 3*x*x*y, which is equal to 3x²y. Now we can complete the factoring of the expression by grouping the remaining terms together.

3x²y(y+3x+2xy)

And we’re done!