Mathematics » Factoring and Factorisation I » Greatest Common Factor and Factor by Grouping

# Finding the Greatest Common Factor of Two Or More Expressions

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

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.

1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. 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 $$3x.$$

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