8.2 - Factoring a difference of squares


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



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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