Algebra I in Simple English/Factoring/Factoring a^2-b^2 Binomials

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Difference of Squares

Any binomial of the form a² − b² may be written as . That is

.

Example 1: Factor x² − 9.

This is clearly seen just take a² = x² and b² = 9 so that b = 3. So x² − 9 = (x − 3)(x + 3)

Example 2:: 32w⁴ − 162.

Here is is unclear where we can use the difference of squares as 32 is NOT a perfect square. However if we look we see that we can factor out a common factor of 2.

32w⁴ − 162 = 2(16w⁴ − 81)

Now we see we can use the difference of two squares to simplify matters take a² = 16w⁴ and b² = 81:

2(16w⁴ − 81) = 2(4w² − 9)(4w² + 9)

Now we notice that we can use the difference of squares again in the first factor to get:

2(4w² − 9)(4w² + 9) = 2(2w + 3)(2w − 3)(4w² + 9)

This is now completely factored.

This is brings us to our next point that is that a² + b² is NOT FACTORABLE (at least for the purposes of this class).