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Finding the Greatest Common Factor

Quick Check - Identifying Polynomials

Find the greatest common factor of the following monomials:

 
 

The first step in learning to factor polynomials is learning to apply the Distributive Property in reverse. When you first learn about the Distributive Property you see it as a(b + c) = ab + ac and that's how you use it - to get rid of parentheses. When we talk about factoring polynomials, we want to use it the other way:

ab + ac = a(b + c)

We want to look for things that the parts of an expression have in common, for example, the a's in the expression on the right. Then to factor the expression, we'll take those common things and pull them outside a set of parentheses, leaving everything else behind, inside the parentheses.

In practice, what we need to do is find the greatest common factor of each of the polynomial's factors. This process breaks down to two steps:

  1. Find the greatest common factor of all the coefficients.
  2. List all of the variables that the terms of the expression have in common. Take the smallest exponent for each variable from the expression's terms.

Multiply the number from step one with the variable expression from step two and you've got the greatest common factor. Let's try a few examples.

Example 1

Find the greatest common factor of 12x and 4xy.

The numbers are usually the easy part to work with so lets start there. The greatest common factor of 12 and 4 is 4 because 4 is the largest number that divides both 12 and 4.

To get the variable part, we need to start by lisiting all the variables that the expressions have in common. In this case, the only variable they have in common is an x. Now we need to look at the two expressions and find the smallest exponent of the x parts. In both of expressions, that's a 1 so the variable part of our greatest common factor will be x1 = x.

Now our final answer is the product of 4 and x or just 4x.

Example 2

Find the greatest common factor of 2ab2 and 7a4b2c3.

Again, we'll start with the coefficients. Since there are no numbers that divide both 2 and 7, the coefficient of our greatest common factor will have to be 1.

For the variable part, the only things that the two expressions have in common are a and b so those are the only variables that will be in our greatest common factor. Since the smallest exponent on the a variable is 1 and the smallest exponent on the b variables is 2, the variable part of our greatest common factor will be ab2.

Putting those two conclusions together leaves us with 1ab2 = ab2 as our greatest common factor.