## 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

EXAMPLE 1

Factor: $45x^2-500$

###### SOLUTION

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

Evaluating: $2^3=8$

EXAMPLE 2

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: $m^0=1$

EXAMPLE 3

Factor: $x^2+49$

###### SOLUTION

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

Therefore: $x=3$

EXAMPLE 4

Factor: $x^4-16$

###### SOLUTION

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

Therefore: $x=3$