The Greatest Common Factor

Learning Outcomes

• Identify the greatest common factor of a polynomial

Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: $2$ and $10$ are factors of $20$, as are $4, 5, 1, 20$. To factor a number is to rewrite it as a product. $20=4\cdot{5}$ or $20=1\cdot{20}$. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product. Factoring is very helpful in simplifying expressions and solving equations involving polynomials.

The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, $4$ is the GCF of $16$ and $20$ because it is the largest number that divides evenly into both $16$ and $20$ The GCF of polynomials works the same way: $4x$ is the GCF of $16x$ and $20{x}^{2}$ because it is the largest polynomial that divides evenly into both $16x$ and $20{x}^{2}$.

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Greatest Common Factor

The greatest common factor (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.

Example

Find the greatest common factor of $25b^{3}$ and $10b^{2}$.

In the example above, the monomials have the factors $5$, b, and b in common, which means their greatest common factor is $5\cdot{b}\cdot{b}$, or simply $5b^{2}$.

The video that follows gives an example of finding the greatest common factor of two monomials with only one variable.

Sometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example, we find the GCF of two terms which both contain two variables.

Example

Find the greatest common factor of $81c^{3}d$ and $45c^{2}d^{2}$.

The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.

Now that you have practiced identifying the GCF of terms with one and two variables, we can apply this idea to factoring the GCF out of a polynomial. Notice that the instructions are now “Factor” instead of “Find the greatest common factor.”

To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in factored form. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.

Distributive Property Forward and Backward

Forward: Product of a number and a sum: $a\left(b+c\right)=a\cdot{b}+a\cdot{c}$. You can say that “$a$ is being distributed over $b+c$.”

Backward: Sum of the products: $a\cdot{b}+a\cdot{c}=a\left(b+c\right)$. Here you can say that “$a$ is being factored out.”

We first learned that we could distribute a factor over a sum or difference, now we are learning that we can “undo” the distributive property with factoring.

Example

Factor $25b^{3}+10b^{2}$.

The factored form of the polynomial $25b^{3}+10b^{2}$ is $5b^{2}\left(5b+2\right)$. You can check this by doing the multiplication. $5b^{2}\left(5b+2\right)=25b^{3}+10b^{2}$.

Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.

For example:

$\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}\end{array}$

Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.

In the following video, we show two more examples of how to find and factor the GCF from binomials.

We will show one last example of finding the GCF of a polynomial with several terms and two variables. No matter how large the polynomial, you can use the same technique described below to factor out its GCF.

How To: Given a polynomial expression, factor out the greatest common factor

1. Identify the GCF of the coefficients.
2. Identify the GCF of the variables.
3. Write together to find the GCF of the expression.
4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Example

Factor $6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy$.

In the following video, you will see two more example of how to find and factor our the greatest common factor of a polynomial.