Greatest Common Factor

In chapter 1, we defined a factor as a number that divides exactly into another number. We multiplied factors together to get a product.  Factors are the building blocks of multiplication. They are the numbers that we can multiply together to produce another number: [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/latex], as are  [latex]4[/latex] and [latex]5[/latex] and [latex]1[/latex] and [latex]20[/latex]. To factor a number is to rewrite it as a product. For example, [latex]20=4\cdot5[/latex]. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – rewriting a sum or difference as a product.

Finding a product is multiplying two or more terms together. Each term is a factor of the product. Splitting a product into factors is called factoring.

A prime factor is a prime number—it has only itself and 1 as factors—that is a factor. The process of breaking a number down into its prime factors is called prime factorization. Prime factorization is unique. Each number has only one set of prime factors. For example, [latex]10=2\cdot 5[/latex] is the only way to write [latex]10[/latex] as a product of primes. To find the GCF, we can factor each number into its prime factors, identify the prime factors the numbers have in common, and then multiply those prime factors together.

example

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

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]

The video that follows shows another example of finding the greatest common factor of two whole numbers.

Greatest Common Factor of Monomials

Determining the GCF of monomials works the same way as numbers: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20x^2[/latex] because it is the largest term that divides exactly into both [latex]16x[/latex] and [latex]20x^2[/latex].

Finding the greatest common factor in a set of monomials is the same as finding the GCF of two whole numbers. The only difference is that there will be variables involved. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.

example

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

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 greatest common factor 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]

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]

example

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

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]

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

Factoring a Polynomial

A polynomial is made up of the sum (or difference) of monomial terms. If all of these monomial terms have a greatest common factor, we can factor the polynomial. Factoring a polynomial is very helpful in simplifying and solving polynomial equations.

Consider the polynomial [latex]10x^4+30x^3-45x^2[/latex]. The monomials that make up this polynomial are [latex]10x^4\text{, }30x^3[/latex] and [latex]-45x^2[/latex]. The GCF of these monomials is [latex]5x^2[/latex]. This means that [latex]5x^2[/latex] divides exactly into each of the monomials.  If we were to divide the polynomial by the GCF we would get:

[latex]\begin{equation}\begin{aligned}& \;\;\;\;\frac{10x^4+30x^3-45x^2}{5x^2} \\ &=\frac{10x^4}{5x^2}+\frac{30x^3}{5x^2}-\frac{45x^2}{5x^2} \\ &=2x^2+6x-9\end{aligned}\end{equation}[/latex]

This means that [latex]5x^2[/latex] and [latex]2x^2+6x-9[/latex] are factors of [latex]10x^4+30x^3-45x^2[/latex].

In other words, [latex]10x^4+30x^3-45x^2=5x^2\left (2x^2+6x-9\right )[/latex].  We have factored the polynomial [latex]10x^4+30x^3-45x^2[/latex] using the greatest common factor.

Note that we can always check our work by using the distributive property to multiply [latex]5x^2\left (2x^2+6x-9\right )[/latex] to get back to the original polynomial [latex]10x^4+30x^3-45x^2[/latex].

Notice that every time, the exponent on the variable in the GCF is always the lowest exponent in the polynomial. This will always be the case.

When the leading coefficient of a polynomial is negative, we include the negative sign as part of the GCF.

As you gain confidence with factoring you will find that you can skip the division step and simply use the distributive property to factor.

The following videos provide more examples of factoring a polynomial using the distributive property.

Sometimes the GCF of the monomials in a polynomial is [latex]1[/latex]. For example, the polynomial [latex]6x^2-7x+2[/latex] has a greatest common factor of [latex]1[/latex] across the monomials making up the polynomial. Factoring the polynomial as [latex]6x^2-7x+2=1\left (6x^2-7x+2\right )[/latex] isn’t of much use. However, we will see in the next sections that there are other ways to factor a polynomial. Indeed, [latex]6x^2-7x+2[/latex] factors into binomial factors [latex](3x-2)(2x-1)[/latex].