# Greatest Common Factor

### Learning Outcomes

- Find the greatest common factor of a list of expressions
- Find the greatest common factor of a polynomial

**Factors**are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/latex], as are [latex]4, 5, 1, 20[/latex]. To factor a number is to rewrite it as a product. [latex]20=4\cdot{5}[/latex] or [latex]20=1\cdot{20}[/latex]. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product.

**Factoring**is very helpful in simplifying expressions and solving equations involving polynomials. The

**greatest common factor**(GCF) of two numbers is the largest number that divides evenly into both numbers: [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex]. The GCF of two or more algebraic expressions 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 algebraic expression that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

### 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.

### try it

[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]. |

### try it

[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] |

### try it

[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] |

### try it

[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].

### try it

[ohm_question]14137[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Question ID 146329, 146328, 146327, 146326, 14137.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution. - Revision and Adaptation.
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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.
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- Prealgebra.
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