When multiplying polynomials, we apply the distributive property:a\left(b + c\right) = ab + acWe can also apply this in reverse, which is known as factoring an expression:xy + xz = x\left(y + z\right)

There can be many steps when factoring a polynomial expression. To begin with, we first want to identify the greatest common factor, or GCF, of the terms in the expression.

Greatest common factor (GCF)

The largest whole number or algebraic expression that evenly divides the given expression.

The GCF of two or more terms includes the largest numeric factor of the coefficients of each term, and the lowest power of any variable that appears in every term. (If a variable does not appear in a term, it can be thought of as if it had an exponent of 0.)

Once an expression has been factored, we can verify the factored form by multiplying. The product should be the original expression.

Worked examples

Example 1

Identify the greatest common factor between 12 x^{5} y^{3} z^{6} and 18 w^{2} x^{7} y z^{4}

Approach

We find the largest whole number that evenly divides the coefficients of 12 x^{5} y^{3} z^{7} and 18 w^{2} x^{4} y z^{3} and find the expression with the lowest power of each of the variables.

Solution

The largest whole number that evenly divides the coefficients 12 and 18 is 6. The expression with the lowest power of each of the variables is x^{5}yz^{4}.

Putting this together, the GCF of this expression is 6x^{5}yz^{4}.

Example 2

Factor the expression 8 x y^{4} z^{3} - 16 x^{3} y^{2} z + 4 x y^{3} z^{5}

Approach

We will first find the GCF of 8 x y^{4} z^{3}, - 16 x^{3} y^{2} z, and 4 x y^{3} z^{5}. We can then rewrite each expression as a product of the GCF and any remaining factors, and then factor out the GCF.

Solution

The GCF of 8 x y^{4} z^{3}, - 16 x^{3} y^{2} z \text{ and } 4 x y^{3} z^{5} is 4xy^{2} z. So we have:

\displaystyle 8 x y^{4} z^{3} - 16 x^{3} y^{2} z + 4 x y^{3} z^{5}	\displaystyle =	\displaystyle 4xy^{2} z\left(2y^{2}z^{2}\right) - 4xy^{2} z\left(4 x^{2}\right) + 4xy^{2} z\left(yz^{4}\right)
	\displaystyle =	\displaystyle 4xy^{2} z\left(2y^{2}z^{2} - 4 x^{2} + yz^{4} \right)

Reflection

We can check the answer by distributing the multiplication:

\displaystyle 4xy^{2} z\left(2y^{2}z^{2} - 4 x^{2} + yz^{4} \right)	\displaystyle =	\displaystyle 4xy^{2} z\left(2y^{2}z^{2}\right) - 4xy^{2} z\left(4 x^{2}\right) + 4xy^{2} z\left(yz^{4}\right)
	\displaystyle =	\displaystyle 8 x y^{4} z^{3} - 16 x^{3} y^{2} z + 4 x y^{3} z^{5}

Notice that, if no mistakes have been made, these are the same steps just in reverse.

Example 3

Factor the expression 3x\left(x-4\right)+7\left(x-4\right)

Approach

This time, the expressions are themselves already factored. We can use this to help identify the GCF.

Solution

In particular, notice that both terms 3x\left(x-4\right) and 7\left(x-4\right) have a factor of \left(x-4\right).

Furthermore the remaining parts of each expression, 3x and 7, have no factors in common. So the GCF is \left(x - 4\right), which we can use to factor the expression:

\displaystyle 3x\left(x-4\right) + 7\left(x-4\right)

\displaystyle =

\displaystyle \left(x-4\right)\left(3x + 7\right)

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.1.7

Rewrite a polynomial expression as a product of polynomials over the real number system.

6.05 Factoring GCF

Concept summary

Worked examples

Example 1

Approach

Solution

Example 2

Approach

Solution

Reflection

Example 3

Approach

Solution

Outcomes

MA.912.AR.1.1

MA.912.AR.1.7

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