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

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