Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/latex], as are [latex]4, 5, 1, 20[/latex]. To factor a number is to rewrite it as a product. [latex]20=4\cdot{5}[/latex] or [latex]20=1\cdot{20}[/latex]. 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: [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex]. The GCF of two or more algebraic expressions works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20{x}^{2}[/latex] because it is the largest algebraic expression that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

Find the GCF of a list of algebraic expressions

We begin by finding the GCF of a list of numbers, then we’ll extend the technique to monomial expressions containing variables.

A good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.

Here’s a summary of the technique.

In the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.

To find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here’s an example using the matching method from the example above.

 

In the examples so far, the GCF was a constant. In the next two examples we will get variables in the greatest common factor.

 

Here are some examples of finding the GCF of a list of more than two expressions.

 

The common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term.

In fact, the GCF of a set of expressions in the form [latex]{x}^{n}[/latex] will always be the exponent of lowest degree.

Watch the following video to see another example of how to find the GCF of two monomials that have 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.

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

Find the GCF of a polynomial

Now that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring the GCF out of a polynomial. That is, doing the distributive property “backwards” to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial.

Recall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions.

No matter how large the polynomial, you can use the same technique described below to identify its GCF.

 

 

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

Finding and factoring out a GCF from a polynomial is the first skill involved in factoring polynomials.

Factoring a GCF out of a polynomial

When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.

To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property “backwards” to rewrite the polynomial in a 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.

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

The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.

Watch this video to see more examples of how to factor the GCF from a trinomial.

Factor a Four Term Polynomial by Grouping Terms

When we learned to multiply two binomials, we found that the result, before combining like terms, was a four term polynomial, as in this example: [latex]\left(x+4\right)\left(x+2\right)=x^{2}+2x+4x+8[/latex].

We can apply what we have learned about factoring out a common monomial to return a four term polynomial to the product of two binomials. Why would we even want to do this?

Because it is an important step in learning techniques for factoring trinomials, such as the one you get when you simplify the product of the two binomials from above:

[latex]\begin{array}{l}\left(x+4\right)\left(x+2\right)\\=x^{2}+2x+4x+8\\=x^2+6x+8\end{array}[/latex]

Additionally, factoring by grouping is a technique that allows us to factor a polynomial whose terms don’t all share a GCF. In the following example, we will introduce you to the technique. Remember, one of the main reasons to factor is because it will help solve polynomial equations.

Notice that when you factor a two term polynomial, the result is a monomial times a polynomial. But the factored form of a four-term polynomial is the product of two binomials. As we noted before, this is an important middle step in learning how to factor a three term polynomial.

This process is called the grouping technique. Broken down into individual steps, here’s how to do it (you can also follow this process in the example below).

• Group the terms of the polynomial into pairs that share a GCF.
• Find the greatest common factor and then use the distributive property to pull out the GCF
• Look for the common binomial between the factored terms
• Factor the common binomial out of the groups, the other factors will make the other binomial

Let’s try factoring a few more four-term polynomials. Note how there is a now a constant in front of the [latex]x^2[/latex] term. We will just consider this another factor when we are finding the GCF.

Another example follows that contains subtraction. Note how we choose a positive GCF from each group of terms, and the negative signs stay.

The video that follows provides another example of factoring by grouping.

In the next example, we will have a GCF that is negative. It is important to pay attention to what happens to the resulting binomial when the GCF is negative.

In the following video we present another example of factoring by grouping when one of the GCF is negative.

Sometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.

In the example above, each pair can be factored, but then there is no common factor between the pairs! This means that it cannot be factored.