8.2 - Difference of Squares

Just as the name suggests, a difference of squares is an expression of the form:

a² − b².

A difference of squares can be factored like this:

a² − b² = (a + b) (a − b).

You can easily verify this by multiplying out the right hand side and noticing that the cross terms cancel:

Here are some examples showing the range of expressions that qualify as a difference of squares.

Example: Factor x² − 4.

The second term can be thought of as the square of the number 2:

Example: Factor x² − 5.

The second term can be thought of as the square of the square root of 5:

Example: Factor x⁶ − 0.64.

The first term can be thought of as the square of x³:

Example: Factor (a + b)² − c².

In this case the first term is the multinomial (a + b):

You should now simplify by dropping the unnecessary brackets and get

Similar to a difference of squares, a sum of squares is an expression of the form:

a² + b².

A sum of squares cannot be factored over the real numbers but over the complex numbers it can be factored like this:

a² + b² = (a + b i ) (a − b i ),

where

You can easily verify this by multiplying out the right hand side and then canceling the cross terms:

and then simplifying the last term to +b² because i² = −1.

Here are some examples showing the range of expressions that qualify as a sum of squares.

Example: Factor x² + 4.

The second term can be thought of as the square of the number 2:

Example: Factor x² + 5.

The second term can be thought of as the square of the square root of 5:

Example: Factor x⁶ + 0.64.

The first term can be thought of as the square of x³.

Each factor can actually be factored further than is shown here. Click here to see how.

Example: Factor (a + b)² + c².

In this case the first term is the multinomial (a + b):

You should now simplify by dropping the unnecessary brackets and get

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