## Advanced Factoring

A **difference of squares** has two terms each of which are **perfect squares**. That is, they can be square rooted “nicely.”

1, 4, 9, 16, 25, ldots are perfect squares, as well as polynomials like x^2, y^2, x^4, y^6, a^2b^4, etc.

There must also be ** minus sign** in between the two terms, hence its name, a difference of squares.

##### How to Factor a Difference of Squares

Add the square root of each term as the first factor. Then, subtract the square root of each term as the second factor.

For example: x^2-16

The square root of the first term is x. The square root of the second term is 4. Therefore:x^2-16=(x+4)(x-4)

### Practice

Factor: 45x^2-500

###### SOLUTION

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

Evaluating: 2^3=8

Factor: 25a^6-36b^4

###### 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

Factor: x^2+49

###### SOLUTION

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

Therefore: x=3

Factor: x^4-16

###### SOLUTION

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

Therefore: x=3

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