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How to Find the Greatest Common Factor of a Polynomial

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

When working algebra problems, you'll constantly find yourself simplifying equations and solving for unknowns. This often involves finding the greatest common factor of a polynomial, which you'll learn how to do here.

Factors

What determines how long it takes you to get to school each day? If you're being driven or taking the bus, the speed limit certainly has some effect on how long it takes to get there. However, so do the number of red lights you hit and how bad the traffic is in the morning. We can say these are all factors in determining how long it takes you to get to school. Factors are the components that contribute to an overall result, and this idea is also found in math.

You'll be working with factors in math classes from when you first learn to multiply all the way through college. Whenever you multiply two or more numbers together to get a new number, those multiplied numbers are factors of the answer you get. For example, 3 and 4 are both factors of 12.


twelve factored


The factors of a number can be broken down until they are all prime numbers - numbers that have no factors other than 1 and the number itself. We call this process prime factorization. In our previous example, 4 can be broken down into two prime factors of 2.


twelve prime factors


Knowing how to find factors and prime factors is very important in algebra. We use this all the time when simplifying algebra problems and solving for unknowns. More specifically, it's finding something called the greatest common factor that will be used frequently when working with polynomials.

GCF of Whole Numbers

You have probably already learned how to find the greatest common factor of whole numbers at some point. You'll also need to know how to do this when working with polynomials, so we'll quickly review it.

A greatest common factor (GCF) is the largest factor shared between two or more numbers. To see what we mean by this, let's find the GCF of 45 and 75. The first step in determining a GCF is to work out the prime factorization of each number.


number prime factorization


Once we've done this, we need to determine which numbers the prime factorizations have in common.


common factors


We can see they both share a 3 and a 5. We find the greatest common factor by multiplying these shared factors together.


GCF answer


So in the end, we determine that the GCF of 45 and 75 is 15.

Polynomials vs. Whole Numbers

We've seen how to find the GCF of whole numbers, but before we dive into finding one for a polynomial, there are a couple differences between the two methods that we need to address.

The first difference is that you have to factor variables along with whole numbers. Variables are factored completely by writing their powers out as that variable multiplied by itself as many times as the power indicates. For example, we would factor x3 as follows:


variable factors


When searching for the GCF of two or more variables, you can break them down into their smallest increments and compare them just like when we find the prime factors of whole numbers, but there is also a faster method. The GCF of any number of variables is always equal to the lowest power shared between them. To see what I mean by this, let's look at the following two monomials.


variables to compare


These monomials share two variables, b and c. The lowest power for b is b2 and the lowest power for c is c3. So, the GCF of these two expressions would be b2c3.

The second difference is that instead of looking for a GCF between multiple polynomials, like we did with the whole numbers, it's more common to be looking for a GCF of a single polynomial. You might be thinking that greatest common factors are comparisons of two or more numbers, so how can we possibly do it for a single polynomial? Well, what we are comparing are the different terms within the polynomial.


Terms of a Trinomial
trinomial terms


Polynomials are expressions made up of terms that consist of numbers and variables multiplied together. The terms that we're factoring are separated by addition and subtraction signs in the polynomial.

Finding the GCF of a Polynomial

We now have everything we need to factor a polynomial. Let's put it all together by finding the GCF of the following example.


example polynomial


We'll be working with three different terms in this polynomial.


polynomial terms


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