Greatest Common Factor
Learning Outcomes
- Find the greatest common factor of a list of expressions
- Find the greatest common factor of a polynomial
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].Answer:
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]. |
[latex]\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}[/latex]
Find 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.
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[ohm_question]146326[/ohm_question]example
Find the greatest common factor of [latex]5x\text{ and }15[/latex].Answer: 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]. |
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[ohm_question]146327[/ohm_question]example
Find the greatest common factor of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex].Answer: 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] |
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[ohm_question]146328[/ohm_question]example
Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[/latex].Answer: 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] |
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[ohm_question]146329[/ohm_question]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}\end{array}[/latex]
Answer
[latex-display]\text{GCF}=9c^{2}d[/latex-display]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.How To: Given a polynomial expression, find the greatest common factor.
- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
Example
Find the GCF of [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].Answer: 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]. And the GCF of [latex]{y}^{3},{y}^{2}[/latex], and [latex]y[/latex] is [latex]y[/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[/latex].
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[ohm_question]14137[/ohm_question]Licenses & Attributions
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- Question ID 146329, 146328, 146327, 146326, 14137. Authored by: Lumen Learning. License: CC BY: Attribution.
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CC licensed content, Shared previously
- Ex: Determine the GCF of Two Monomials (One Variables). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex: Determine the GCF of Two Monomials (Two Variables). 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.
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- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].