The Greatest Common Factor
Learning Outcomes
- Identify the greatest common factor of a polynomial
Greatest Common Factor
The greatest common factor (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.Example
Find the greatest common factor of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex].Answer:
[latex]\begin{array}{l}\,\,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}\\\text{GCF}=5b^{2}\end{array}[/latex]
Example
Find the greatest common factor of [latex]81c^{3}d[/latex] and [latex]45c^{2}d^{2}[/latex].Answer:
[latex]\begin{array}{l}\,\,\,81c^{3}d=3\cdot3\cdot3\cdot3\cdot{c}\cdot{c}\cdot{c}\cdot{d}\\45c^{2}d^{2}=3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}\\\,\,\,\,\text{GCF}=3\cdot3\cdot{c}\cdot{c}\cdot{d}\\\,\,\,\,\text{GCF}=9c^{2}d\end{array}[/latex]
Distributive Property Forward and Backward
Forward: Product of a number and a sum: [latex]a\left(b+c\right)=a\cdot{b}+a\cdot{c}[/latex]. You can say that “[latex]a[/latex] is being distributed over [latex]b+c[/latex].” Backward: Sum of the products: [latex]a\cdot{b}+a\cdot{c}=a\left(b+c\right)[/latex]. Here you can say that “[latex]a[/latex] is being factored out.” We first learned that we could distribute a factor over a sum or difference, now we are learning that we can "undo" the distributive property with factoring.Example
Factor [latex]25b^{3}+10b^{2}[/latex].Answer: Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex] to be [latex]5b^{2}[/latex].
[latex]\begin{array}{l}\,\,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}=5b^{2}\end{array}[/latex]
Rewrite each term with the GCF as one factor.[latex]\begin{array}{l}25b^{3} = 5b^{2}\cdot5b\\10b^{2}=5b^{2}\cdot2\end{array}[/latex]
Rewrite the polynomial using the factored terms in place of the original terms.[latex]5b^{2}\left(5b\right)+5b^{2}\left(2\right)[/latex]
Factor out the [latex]5b^{2}[/latex].[latex]5b^{2}\left(5b+2\right)[/latex]
[latex]\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}\end{array}[/latex]
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually. In the following video, we show two more examples of how to find and factor the GCF from binomials. https://youtu.be/25_f_mVab_4 We will show one last example of finding the GCF of a polynomial with several terms and two variables. No matter how large the polynomial, you can use the same technique described below to factor out its GCF.How To: Given a polynomial expression, factor out the greatest common factor
- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Write together to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
Example
Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].Answer: First, find the GCF of the expression. The GCF of [latex]6,45[/latex], and [latex]21[/latex] is [latex]3[/latex]. The GCF of [latex]{x}^{3},{x}^{2}[/latex], and [latex]x[/latex] is [latex]x[/latex]. (Note that the GCF of a set of expressions in the form [latex]{x}^{n}[/latex] will always be the exponent of lowest degree.) And the GCF of [latex]{y}^{3},{y}^{2}[/latex], and [latex]y[/latex] is [latex]y[/latex]. Put these together to find the GCF of the polynomial, [latex]3xy[/latex]. Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3},3xy\left(15xy\right)=45{x}^{2}{y}^{2}[/latex], and [latex]3xy\left(7\right)=21xy[/latex]. Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by [latex-display]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)[/latex-display] After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 1: Identify GCF and Factor a Binomial. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Ex 2: Identify GCF and Factor a Trinomial. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.