Greatest Common Factor

A common factor between a set of terms is a polynomial that will divide into each term evenly or “nicely”.

For example, $2x$ is a common factor of $8x^2$ and $20x^3$ because $\frac{8x^2}{2x}=4x$ and $\frac{20x^3}{2x}=10x^2$.

$4$, $x$, $x^2$ are also common factors of $8x^2$ and $20x^3$.

The greatest common factor is the common factor with the largest coefficient and degree.

For example, the common factors of $8x^2$ and $20x^3$ are $1, 2, 4, x, 2x, 4x, x^2, 2x^2, 4x^2$. Therefore, $4x^2$ is the greatest common factor.

EXAMPLE: Determine the greatest common factor:

a) ${6,8}$

GCF = $2$

b) ${12,16,20}$

GCF = $4$

c) ${5,8}$

GCF = $1$

d) ${2x,3x}$

GCF = $x$

e) ${4x^2, 2x, 2}$

GCF = $2$

f) ${6x^3y^2,18x^2yz,24x^2z^2}$

GCF = $6x^2$

Practice

EXAMPLE 1

Simplify then evaluate: $2^{11} div 2^8$

SOLUTION

Using the Law of Division: $$2^{11} div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Simplify: $(m^0)^{10}$

SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Find the value of $x$ that makes the equation true: $4^x times 4^4=4^7$

SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$