## Finding the Greatest Common Factor of Two Or More Expressions

Contents

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called **factoring**.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the **greatest common factor** of two or more expressions. The method we use is similar to what we used to find the LCM.

### Greatest Common Factor

The **greatest common factor** (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

First we’ll find the GCF of two numbers.

### Example: How to Find the Greatest Common Factor of Two or More Expressions

Find the GCF of 54 and 36.

### Solution

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

\(\begin{array}{c}54=18·3\hfill \\ 36=18·2\hfill \end{array}\)

We summarize the steps we use to find the GCF below.

### Find the Greatest Common Factor (GCF) of two expressions.

- 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.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

## Example

Find the greatest common factor of \(27{x}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}18{x}^{4}\).

### 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 \(27{x}^{3}\) and \(18{x}^{4}\) is \(9{x}^{3}.\) |

## Example

Find the GCF of \(4{x}^{2}y,6x{y}^{3}\).

### 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 \(4{x}^{2}y\) and \(6x{y}^{3}\) is \(2\mathrm{xy}\). |

## Example

Find the GCF of: \(21{x}^{3},9{x}^{2},15x\).

### 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 \(21{x}^{3}\), \(9{x}^{2}\) and \(15x\) is \(3x.\) |

Great tutorial lesson. it is quiet simple to understand and am happy and greatfull for the fact that i now know how to find the greatest common factor of two or more Expressions.