## 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

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

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

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

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