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6. Polynomials and Factoring
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Algebra I

6.04 Factoring GCF

Lesson
Practice

Introduction

We identified the greatest common factor between two whole numbers in 6th grade and rewriting the sum of two numbers with a common factor as the sum of two numbers without a common factor. In this lesson, we will extend this concept to factoring polynomial expressions.

Identifying a GCF

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

Examples

Example 1

Find the greatest common factor of the given terms.

a

60 and 24.

Worked Solution
Create a strategy

List the prime factorization of 60 and 24, then determine the common factors that comprise the GCF.

Apply the idea

The prime factorization of 60 is 2 \cdot 2 \cdot 3 \cdot 5

The prime factorization of 24 is 2 \cdot 2 \cdot 2 \cdot 3.

The GCF is the product of the common factors: 2 \cdot 2 \cdot 3 = 12.

Therefore, the GCF of 60 and 24 is 12.

Reflect and check

Instead of working through the entire prime factorization, we can list the whole number factors of 60 and 24, then determine which common factor is the largest.

The whole number factors of 60 are 1, 60, 2, 30, 3, 20, 4, 15, 5, 12, 6, 10.

The whole number factors of 24 are 1, 24, 2, 12, 3, 8, 4, 6.

Based on the list of whole number factors, 12 is the GCF of 60 and 24.

b

60x^3y^2 and 24xy^4.

Worked Solution
Create a strategy

List the whole number factors of the coefficients of 60x^3y^2 and 24xy^4 and find the expression with the lowest power of each of the variables.

Apply the idea

We know that the largest whole number that 60 and 24 are divisible by is 12. The expression with the lowest power of each of the variables is xy^2.

Putting this together, the greatest common factor is 12xy^2.

Example 2

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

Worked Solution
Apply the idea

The expression with the lowest power of each of the variables is x^{5}yz^{4}.

Therefore, the GCF of x^{5} y^{3} z^{6} and w^{2} x^{7} y z^{4} is x^{5}yz^{4}.

Reflect and check

Another way to find the GCF of the monomials is by expanding the terms and determining the highest number of common factors.

x^5y^3z^6=x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot z \cdot z \cdot z \cdot z \\ w^{2} x^{7} y z^{4}= w \cdot w \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot z \cdot z \cdot z \cdot z

There are no common factors of w between the terms. There are 5 common x factors, 1 common y, and 4 common z factors.

Idea summary

The GCF of monomials is the expression with the lowest power of each of the variables from each of the terms.

Factoring out the GCF

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

Given a polynomial expression, we can factor out a GCF. The process is the opposite of distribution. Follow these steps for factoring out a GCF:

  1. Identify the GCF
  2. Rewrite each term as a product of the GCF and the remaining factors
  3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses

Examples

Example 3

Factor the expression 8x^2 + 4x.

Worked Solution
Apply the idea

The GCF of 8x^2 and 4x is 4x. So we have:

\displaystyle 8x^2+4x\displaystyle =\displaystyle 4x(2x) + 4x(1)
\displaystyle =\displaystyle 4x(2x+1)
Reflect and check

Although the term 4x is in the original expression when it is factored out the second term does not become zero. Otherwise, when we check the answer by distributing the multiplication, 4x will be lost altogether.

Example 4

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

Worked Solution
Create a strategy

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

Apply the idea

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)

Example 5

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

Worked Solution
Create a strategy

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.

Apply the idea

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 will divide each term by 4xy^{2} z:

\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(\dfrac{8 x y^{4} z^{3}}{4xy^{2} z} - \dfrac{16 x^{3} y^{2} z}{4xy^{2} z} + \dfrac{4 x y^{3} z^{5}}{4xy^{2} z}\right)
\displaystyle =\displaystyle 4xy^{2} z\left(\dfrac{2 x y^{4} z^{3}}{xy^{2} z} - \dfrac{4 x^{3} y^{2} z}{xy^{2} z} + \dfrac{ x y^{3} z^{5}}{xy^{2} z}\right)Divide the coefficients
\displaystyle =\displaystyle 4xy^{2} z\left(2y^{2}z^{2} - 4 x^{2} + yz^{4} \right)Quotient of powers
Reflect and check

We can check the answer by distributing the multiplication and using the product of powers rule for exponents:

\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.

Idea summary

Follow these steps for factoring out a GCF:

  1. Identify the GCF
  2. Rewrite each term as a product of the GCF and the remaining factors
  3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses

Outcomes

A.2c

Perform operations on polynomials, including factoring completely first- and second-degree binomials and trinomials in one variable

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