### 8.2 - Factoring a difference of squares

 Just as the name suggests, a difference of squares is an expression of the form: a 2 − b 2. A difference of squares can be factored like this: a 2 − b 2 = (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 x 2 − 4.

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

Example: Factor x 2 − 5.

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

Example: Factor x 6 − 0.64.

The first term can be thought of as the square of x 3:

Example: Factor (a + b) 2c 2.

In this case the first term is the multinomial (a + b):
You should now simplify by dropping the unnecessary brackets and get .

 Algebra Coach Exercises

### Factoring a sum of squares over the complex numbers

 Similar to a difference of squares, a sum of squares is an expression of the form: a 2 + b 2. A sum of squares cannot be factored over the real numbers but over the complex numbers it can be factored like this: a 2 + b 2 = (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 +b 2 because i 2 = −1.

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

Example: Factor x 2 + 4.

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

Example: Factor x 2 + 5.

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

Example: Factor x 6 + 0.64.

The first term can be thought of as the square of x 3.
Each factor can actually be factored further than is shown here. Click here to see how.

Example: Factor (a + b) 2 + c 2.

In this case the first term is the multinomial (a + b):
You should now simplify by dropping the unnecessary brackets and get .

 Algebra Coach Exercises

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