Learning Outcomes
- Determine the greatest common factor of multiple numbers
- Determine the greatest common factor of monomials
- Factor a polynomial using the GCF
Key words
- Factor: a divisor
- Common factor: a term that is a factor of two or more other terms
- Greatest common factor: the common factor that has the highest degree
- Factoring: writing a product or a sum as the multiplication of factors
- Prime number: a whole number greater than or equal to 2 that has exactly 2 factors; 1 and itself
- Prime factor: a factor that is also a prime number
- Prime factorization: the process of factoring a number into prime factors
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.
Example
Determine the greatest common factor of 18 and 24.
Solution
The factors of 18 are 1, 2, 3, 6, 9, and 18
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24
The common factors are the numbers that appear in both lists: 1, 2, 3, and 6.
The greatest common factor is the largest of the common factors: 6.
Answer
GCF(18, 24) = 6
Try It
Determine the greatest common factor of:
- 28 and 42
- 10 and 55
- 64, 16, and 40
- 7, 12, 15
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]
Example
Find the greatest common factor of [latex]210[/latex] and [latex]168[/latex].
Solution
[latex]\begin{array}{l}\,\,\,\,210=2\cdot3\cdot5\cdot7\\\,\,\,\,168=2\cdot2\cdot2\cdot3\cdot7\\\text{GCF}=2\cdot3\cdot7\end{array}[/latex]
Answer
[latex]\text{GCF}=42[/latex]
The video that follows shows another example of finding the greatest common factor of two whole numbers.
try it
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].
Greatest Common Factor of monomials
The greatest common factor (GCF) of two or more monomials is the largest term that is a factor of all the monomials.
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.
determining the greatest common factor
- Factor each coefficient into primes. Write all variables with exponents in expanded form.
- List all factors—matching common factors in a column. In each column, circle the common factors.
- Bring down the common factors that all expressions share.
- Multiply the factors.
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]. |
try it
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]25b^{3}[/latex] and [latex]10b^{2}[/latex].
Solution
[latex]\begin{equation}\begin{aligned}& 25b^{3}&=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\10b^{2}&=5\cdot2\cdot{b}\cdot{b}\\ \text{GCF} &=5\cdot{b}\cdot{b}\end{aligned}\end{equation}[/latex]
The monomials have the factors [latex]5[/latex], b, and b in common, which means their greatest common factor is [latex]5\cdot{b}\cdot{b}[/latex], or simply [latex]5b^{2}[/latex].
Answer
[latex]\text{GCF}=5b^{2}[/latex]
try it
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] |
try it
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].
Example
Factor the polynomial [latex]16x^7+24x^5-56x^3[/latex] using the GCF.
Solution
Find the GCF of the monomials that make up the polynomial:
[latex]16x^7=2\cdot 2\cdot 2\cdot 2 \cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]24x^5=2\cdot 2\cdot 2\cdot 3\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]56x^3=2\cdot 2\cdot 2\cdot 7 \cdot x\cdot x\cdot x[/latex]
GCF = [latex]2\cdot 2\cdot 2\cdot x\cdot x\cdot x=8x^3[/latex]
Divide the polynomial by the GCF:
[latex]\begin{equation}\begin{aligned}&\;\;\;\;\frac{16x^7+24x^5-56x^3}{8x^3}\\&=\frac{16x^7}{8x^3}+\frac{24x^5}{8x^3}-\frac{56x^3}{8x^3}\\&=2x^4+3x^2-7\end{aligned}\end{equation}[/latex]
Write the polynomial as the product of factors:
[latex]16x^7+24x^5-56x^3=8x^3\left (2x^4+3x^2-7\right )[/latex]
Example
Factor the polynomial [latex]14x^6+21x^5-63x^4-77x^3[/latex] using the GCF.
Solution
Find the GCF of the monomials that make up the polynomial:
[latex]14x^6=2\cdot 7\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]21x^5=3\cdot 7\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]63x^4=3\cdot 3\cdot 7\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]77x^3=7\cdot 11\cdot x\cdot x\cdot x[/latex]
GCF=[latex]7x^3[/latex]
Divide the polynomial by the GCF:
[latex]\begin{equation}\begin{aligned}&\;\;\;\;\frac{14x^6+21x^5-63x^4-77x^3}{7x^3}\\&=\frac{14x^6}{7x^3}+\frac{21x^5}{7x^3}-\frac{63x^4}{7x^3}-\frac{77x^3}{7x^3}\\&=2x^3+3x^2-9x-11\end{aligned}\end{equation}[/latex]
Write the polynomial as the product of factors:
[latex]14x^6+21x^5-63x^4-77x^3=7x^3\left (2x^3+3x^2-9x-11\right )[/latex]
Try It
Factor the polynomial [latex]12x^4-21x^3+15x^2[/latex] using the GCF.
Try It
Factor the polynomial [latex]18x^8+54x^5-27x^4+45x^3[/latex] using the GCF.
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.
Example
Factor the polynomial [latex]-5x^6+15x^5-35x^4+45x^3[/latex] using the GCF.
Solution
Find the GCF of the monomials that make up the polynomial:
[latex]5x^6=5\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]15x^5=3\cdot 5\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]35x^4=5\cdot 7\cdot x\cdot x\cdot x\cdot x[/latex]
[latex]45x^3=3\cdot 3\cdot 5\cdot x\cdot x\cdot x[/latex]
GCF = [latex]5x^3[/latex]
Divide the polynomial by the GCF and include the negative sign of the leading coefficient:
[latex]\begin{equation}\begin{aligned}&\;\;\;\;\frac{-5x^6+15x^5-35x^4+45x^3}{-5x^3}\\&=\frac{-5x^6}{-5x^3}+\frac{15x^5}{-5x^3}-\frac{35x^4}{-5x^3}+\frac{45x^3}{-5x^3}\\&=x^3-3x^2+7x-9\end{aligned}\end{equation}[/latex]
Write the polynomial as the product of factors:
[latex]-5x^6+15x^5-35x^4+45x^3=-5x^3\left (x^3-3x^2+7x-9\right )[/latex]
Try It
Factor the polynomial [latex]-6x^6+18x^5-24x^4[/latex] using 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.
Example
Factor the polynomial [latex]18x^3-12x^2[/latex] using the GCF.
Solution
The GCF of the monomials is 2.
So, [latex]18x^3-12x^4=2(\text{a binomial})[/latex] and we need to find that trinomial.
We ask ourselves, [latex]2[/latex] times what equals [latex]18x^3[/latex]? [latex]2(9x^3)=18x^3[/latex] so the first term of the binomial is [latex]9x^3[/latex].
[latex]18x^3-12x^2=2(9x^3+...)[/latex]
Now we ask, what times[latex]2[/latex] equals [latex]-12x^2[/latex]? Well, [latex]2(-6x^2)=-12x^2[/latex], so the second term of the binomial is [latex]-6x^2[/latex].
[latex]18x^3-12x^2=2(9x^3-6x^2)[/latex]
Answer
[latex]18x^3-12x^2=2(9x^3-6x^2)[/latex]TRY IT
try it
The following videos provide more examples of factoring a polynomial using the distributive property.
Try It
Factor the polynomial [latex]14x^4-21x^3-7x^2[/latex] using the GCF.
Try It
Factor the polynomial [latex]-6x^4+21x^3-18x^2[/latex] using the GCF.
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].
