GCF for Exponents

Factoring the GCF

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What is the GCF?

GCF stands for Greatest Common Factor. It is the largest number that goes into the numbers in question.

For example: The GCF of 24 and 36 is 12. That's because 12 is the largest number that goes into 24 and 36 evenly.

Multiple Choice

What is the GCF of 16 and 20?

• 2

• 4

• 5

• 16

Multiple Choice

What is the GCF of 18, 27, and 36?

• 2

• 3

• 9

• 18

The GCF of monomials with variables.

Sometimes we have to find the GCF of monomials with variables when we are factoring.

In order for a variable to be part of the GCF, all numbers must include that variable. Then we take the smallest exponent.

For example: The GCF of 2x³ and 6x² would be 2x².

The GCF for 2 and 6 is 2. The GCF for x³ and x² would be x² because the exponent 2 is smaller than 3.

Multiple Choice

What is the GCF of 12x², 24x, and 6?

• 6

• 6x

• 6x²

• 12x²

Multiple Choice

What is the GCF of 6x³, 12x², and 18x²?

• 3x²

• 6

• 6x

• 6x²

What is factoring?

Factoring is rewriting a polynomial as a product of it's factors.

Factoring out the GCF of a polynomial.

To factor out a GCF from a polynomial we do the following:

1. Find the GCF.

2. Divide the polynomial by the GCF.

3. Rewrite the polynomial as the product of the GCF and the quotient we got when we divided the polynomial by the GCF.

4. Check your work by multiplying.

Example of Factoring out the GCF.

Another Example:

One more example:

Multiple Choice

$$Factor\ \ 5a^4-10a^3+25a^2$$  

• $$5\left(a^4-2a^3+5a^2\right)$$  

• $$5a\left(a^3-2a^2+5a\right)$$  

• $$5a^2\left(a^2-2a+5\right)$$  

• $$5a^3\left(a-2+25a^2\right)$$  

Factoring the GCF with more than 1 variable.

Multiple Choice

$$Factor\ \ \ \ 7x^5y^4-49x^3y^3+21x^2y$$  

• $$7x^2y\left(x^3y^3-7xy^2+3\right)$$  

• $$7xy\left(x^4y^3-7x^2y^2+3x^2\right)$$  

• $$7x^2y^2\left(x^3y^2-7xy+3x^2\right)$$  

• Prime