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.

example

Find the greatest common factor of [latex]24[/latex] and [latex]36[/latex].

Show Solution

Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor [latex]24[/latex] and [latex]36[/latex].
Step 2: List all factors–matching common factors in a column.
In each column, circle the common factors.	Circle the [latex]2, 2[/latex], and [latex]3[/latex] that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.	Bring down the [latex]2, 2, 3[/latex] and then multiply.
Step 4: Multiply the factors.		The GCF of [latex]24[/latex] and [latex]36[/latex] is [latex]12[/latex].

Notice that since the GCF is a factor of both numbers, [latex]24[/latex] and [latex]36[/latex] can be written as multiples of [latex]12[/latex].

[latex]\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}[/latex]

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.

example

Find the greatest common factor of [latex]5x\text{ and }15[/latex].

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Solution

Factor each number into primes. Circle the common factors in each column. Bring down the common factors.
	The GCF of [latex]5x[/latex] and [latex]15[/latex] is [latex]5[/latex].

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

example

Find the greatest common factor of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex].

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Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	The GCF of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex] is [latex]6{x}^{2}[/latex]

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

example

Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[/latex].

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Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	The GCF of [latex]14{x}^{3}[/latex] and [latex]8{x}^{2}[/latex] and [latex]10x[/latex] is [latex]2x[/latex]

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.