6.04 Factor 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.

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.

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 and will reverse polynomial multiplication.

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 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

We can also create factor trees for 60 and 24, then identify the common factors.

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

Therefore, the GCF of 60 and 24 is 12.

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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.

Reflect and check

We can also expand both expressions to find the common factors:

So, the GCF is 2 \cdot 2 \cdot 3 \cdot x \cdot y \cdot y = 12xy^2.

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Example 2

Factor the expression 8x^2 + 4x.

Worked Solution

Create a strategy

Find the GCF and divide it out of each term.

Apply the idea

The GCF of 8x^2 and 4x is 4x.

Dividing out the GCF, we get:

8x^2 \div 4x = 2x

4x \div 4x = 1

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.

We can check our factorization using the distributive property:

4x(2x+1) = 4x(2x) + 4x(1) = 8x^2 +4x

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Example 3

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

Worked Solution

Create a strategy

This time, the terms 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).

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 \dfrac{3x\left(x-4\right) + 7\left(x-4\right)}{x-4}
	\displaystyle =	\displaystyle (x-4)\left(3x \cdot \cancel{\dfrac{x-4}{x-4}} + 7 \cdot \cancel{\dfrac{x-4}{x-4}}\right)
	\displaystyle =	\displaystyle \left(x-4\right)\left(3x + 7\right)

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