Learning Outcomes
- Describe the meaning of greatest common factor
- Determine the greatest common factor of multiple terms in a polynomial
- Factor a polynomial function using GCF method
- Factor a polynomial by grouping
Factors and the Greatest Common Factor
A factor of a number is any number that divides exactly into the number. For example, only the numbers 1, 2, 4, 5, 10, and 20 all divide exactly into 20 with no remainder. Consequently, the factors of 20 are 1, 2, 4, 5, 10, and 20. A factor of an algebraic term is any number, variable, or the combination of number and variable that divides exactly into the term. For example, [latex]6x^2[/latex] is divisible by [latex]1,\,2,\,3,\,6,\,x,\,2x,\,3x,\,6x,\,x^2,\,2x^2,\,3x^2,[/latex] and [latex]6x^2[/latex]. So each of these are factors of [latex]6x^2[/latex].
To factor a term is to rewrite it as a product of factors. For example, [latex]20=4\cdot{5}[/latex] or [latex]6x^2=2x\cdot 3x[/latex]. A prime factor is any factor that is a prime number. When a number is factored into a product of primes, it is called prime factorization. To write a number using prime factorization, we continue to factor it till all of the factors are prime numbers.
Example 1
Determine the prime factorization of:
1. 20
2. 60
3. 90
Solution
| 1.
[latex]\begin{aligned}20&=4\cdot5\\&=2^2\cdot5\end{aligned}[/latex]
|
2.
[latex]\begin{aligned}60&=20\cdot3\\&=4\cdot5\cdot3\\&=2^2\cdot3\cdot5\end{aligned}[/latex] |
3.
[latex]\begin{aligned}90&=9\cdot10\\&=3^2\cdot2\cdot5\\&=2\cdot3^2\cdot5\end{aligned}[/latex] |
If an expression has variables, only the numbers can be factored to a product of primes. The variable(s) is(are) written with an exponent.
Example 2
Write the expression using prime factorization:
1. [latex]6x^2[/latex]
2. [latex]124x^3[/latex]
Solution
| 1.
[latex]\begin{aligned}6x^2&=2\cdot3\cdot x^2\end{aligned}[/latex] |
2.
[latex]\begin{aligned}124x^3&=2\cdot62\cdot x^3\\&=2\cdot2\cdot31\cdot x^3\\&=2^2\cdot31\cdot x^3\end{aligned}[/latex] |
Try It 1
Write the expression using prime factorization:
1. [latex]12[/latex]
2. [latex]42[/latex]
3. [latex]50x^6[/latex]
4. [latex]32x^4[/latex]
In algebra, we use the word factor as both a noun ([latex]2x[/latex] is a factor of [latex]6x^2[/latex]) and as a verb (factor [latex]6x^2[/latex], i.e. [latex]6x^2=2\cdot 3\cdot x\cdot x[/latex]). Factoring can also turn addition or subtraction into a product and is very helpful in simplifying polynomial functions and solving equations involving polynomial functions.
The greatest common factor (GCF) of two numbers is the largest number that divides exactly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides exactly into both [latex]16[/latex] and [latex]20[/latex]. The GCF of terms 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 exactly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].
When factoring a polynomial function, our first step should always be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables. The GCF of the variables will always have the smallest exponent of each variable.
Greatest Common Factor
The greatest common factor (GCF) of a group of terms is the largest term that divides exactly into the polynomials.
We can use prime factorization to find the GCF.
Example 3
Find the greatest common factor of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex].
Solution
[latex]\begin{array}{l}\,\,25b^{3}=\color{blue}{5}\cdot5\cdot{\color{blue}{b}}\cdot{\color{blue}{b}}\cdot{b}\\\,\,10b^{2}=2\cdot\color{blue}{5}\cdot{\color{blue}{b}}\cdot{\color{blue}{b}}\\\text{GCF}=\color{blue}{5}\cdot{\color{blue}{b}}\cdot{\color{blue}{b}}\;\;\;\;\text{Choose factors that appear in each prime factorization.}\\\text{GCF}=5b^{2}\end{array}[/latex]
[latex]25b^3[/latex] and [latex]10b^2[/latex]have the factors [latex]5,\;b[/latex], and [latex]b[/latex] in common, which means their greatest common factor is [latex]5\cdot{b}\cdot{b}=5b^{2}[/latex].
Sometimes we 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 each contain two variables.
Example 4
Find the greatest common factor of [latex]81c^{3}d[/latex] and [latex]45c^{2}d^{2}[/latex].
Solution
Look for factors that are common to each term:
[latex]\begin{array}{l}\,\,\,81c^3d=\color{blue}{3}\cdot\color{blue}{3}\cdot3\cdot3\cdot{\color{blue}{c}}\cdot{\color{blue}{c}}\cdot{c}\cdot{\color{blue}{d}}\\45c^{2}d^{2}=\color{blue}{3}\cdot\color{blue}{3}\cdot5\cdot{\color{blue}{c}}\cdot{\color{blue}{c}}\cdot{\color{blue}{d}}\cdot{d}\\\,\,\,\,\text{GCF}=\color{blue}{3}\cdot\color{blue}{3}\cdot{\color{blue}{c}}\cdot{\color{blue}{c}}\cdot{\color{blue}{d}}\\\,\,\,\,\text{GCF}=9c^{2}d\end{array}[/latex]
Try It 2
Determine the greatest common factor of the terms:
1. [latex]14x^2y^3[/latex]; [latex]21x^3y^2[/latex]
2. [latex]18x^2y[/latex]; [latex]27xy[/latex]
3. [latex]24x^2y^5[/latex]; [latex]36x^2y^3[/latex]; [latex]60x^3y^4[/latex]
Write a Function in Factored Form using the GCF
Now that we 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 function. Notice that the instructions are now “Factor” instead of “Find the greatest common factor.”
To write a polynomial function in factored form, first identify the greatest common factor of the terms. We can then use the distributive property to rewrite the polynomial in factored form. Recall that the distributive property of multiplication over addition states that [latex]a(b+c)=ab+ac[/latex] for all terms [latex]a,\;b,[/latex] and [latex]c[/latex].
Example 5
Write the following function in factored form: [latex]p(b)=25b^{3}+10b^{2}[/latex].
Solution
The GCF of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex] is [latex]5b^{2}[/latex].
Use the distributive property to “pull out” the GCF from all terms in the function by first rewriting each term with the GCF as a factor:
[latex]\begin{aligned}p(b)&=25b^3+10b^2\\ &=\color{blue}{5b^2}\cdot 5b+\color{blue}{5b^2}\cdot 2\\ &=\color{blue}{5b^2} \left (5b+2\right )\end{aligned}[/latex]
The factored form of the polynomial [latex]25b^{3}+10b^{2}[/latex] is [latex]5b^{2}\left(5b+2\right)[/latex]. We can check this by distributing [latex]5b^2[/latex]: [latex]5b^{2}\left(5b+2\right)=25b^{3}+10b^{2}[/latex].
Example 6
Write the following function in factored form: [latex]f(x)=12x^3-16x^2+20x[/latex].
Solution
The GCF of [latex]12x^3,\;16x^2\;,20x=4x[/latex].
Write each term in the function with the GCF as a factor, then “pull out” the GCF using the distributive property.
[latex]\begin{aligned}f(x)&=12x^3-16x^2+20x\\&=\color{blue}{4x}\cdot 3x^2-\color{blue}{4x}\cdot 4x+\color{blue}{4x}\cdot 5\\&=\color{blue}{4x}\left (3x^2-4x+5\right )\end{aligned}[/latex]
Try It 3
Write the following functions in factored form:
1. [latex]f(x)=9x^4-36x^3+45x^2[/latex]
2. [latex]g(x)=15x^5-30x^3-45x^2[/latex]
3. [latex]p(x)=18x^7-36x^5+54x^3[/latex]
When the leading coefficient of a polynomial function is negative, we include the negative sign with the GCF.
Example 7
Write the following function in factored form: [latex]p(x)=-14x^3+21x^2[/latex].
Solution
The GCF = [latex]7x^2[/latex] but we “pull out” [latex]-7x^2[/latex] since the leading coefficient is negative.
Write each term in [latex]p(x)[/latex] using [latex]-7x^2[/latex] as a factor:
[latex]\begin{aligned}-14x^3&=-7x^2\cdot 2x\\ \\21x^2&=-7x^2\cdot (-3)\end{aligned}[/latex]
So, [latex]p(x)=-7x^2\left (2x-3\right )[/latex]
Try It 4
Write the following function in factored form: [latex]p(x)=-8x^4+6x^3-10x^2[/latex]
Write a Polynomial Function in Factored Form using Grouping
When it appears that there is no common factor, except 1, among the terms of a polynomial function, there may be a common factor that is embedded in the polynomial and cannot be seen from the surface. For example, the polynomial function [latex]f(x)=5x^3-20x^2+3x-12[/latex] has no common factor (except 1) across its four terms. However, if we group the first two terms and the last two terms of the polynomial, each pair has a common factor. There is a common factor [latex]5x^2[/latex] in the first group and a common factor [latex]3[/latex] in the second group.
[latex]\begin{aligned}f(x)&=\color{blue}{5x^3-20x^2}+\color{green}{3x-12}\\ \\&=\color{blue}{5x^2}\color{red}{(x-4)}+\color{green}{3}\color{red}{(x-4)}\\ \\&=\color{red}{(x-4)}\left (\color{blue}{5x^2}+\color{green}{3}\right )\end{aligned}[/latex]
If we factor out these common factors from each group, It turns out that we find a common factor [latex]\color{red}{(x-4)}[/latex] across the two groups. After we factor out this common factor, we obtain the factored form [latex]f(x)=(x-4)(5x^2+3)[/latex].
This method of factoring is called factoring by grouping. It does not work for every polynomial, but it can be very useful when it does.
Example 8
Write the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[/latex].
Solution
There is no common factor (other than 1) for all four terms of the function, so we will try factoring by grouping.
[latex]\begin{aligned}g(x)&=5\color{blue}{a^2}x+2\color{blue}{a^2}y\color{green}{-}5\color{green}{b}x\color{green}{-}2\color{green}{b}y\\ \\&=\color{blue}{a^2}\color{red}{(5x+2y)}\color{green}{-b}\color{red}{(5x+2y)}\\ \\&=\color{red}{(5x+2y)}(\color{blue}{a^2}\color{green}{-b})\end{aligned}[/latex]
Notice that since the third term is negative, we factor out the negative sign along with [latex]b[/latex].
Try It 5
Write the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[/latex].
It is not necessary to always group the first two terms and the last two terms. In the next example, we will group the first and third terms, and the second and fourth terms. The next example has the same function as the last example, but we will group the terms differently.
Example 9
Write the following function in factored form: [latex]g(x)=5a^2x+2a^2y-5bx-2by[/latex].
Solution
[latex]\begin{aligned}g(x)&=5a^2x+2a^2y-5bx-2by\\ \\&=5a^2x-5bx+2a^2y-2by\\ \\&=5x(a^2-b)+2y(a^2-b)\\ \\&=(a^2-b)(5x+2y)\end{aligned}[/latex]
Notice that this is the same answer as the previous example with the factors in reverse order.
Try It 6
Write the following function in factored form: [latex]f(x)=4ax^2+7a^2cx-4bx-7abc[/latex] by grouping the 1st and 3rd terms, and the 2nd and 4th terms.
