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The Third Algebraic Identity


You learned what (a + b)(a + b) and (a b)(a b) can be expanded to, but what about (a + b)(a b)? (a + b)(a b) = a2 ab + ba b2 = a2 b2

Rule

The Third Algebraic Identity

(a + b)(a b) = a2 b2

Example 1

Expand (x + 1) (x 1)

If you use the third algebraic identity, you get (x + 1) (x 1) = (x)2 12 = x 1

Sometimes you need to rewrite the expression to be able to use the algebraic identities easily. Look at the following example.

Example 2

Expand (4x 2x2) (x + 2)

You can expand this directly by simply multiplying the parentheses together like you did before. But if you want to be a bit more clever about it, you can rewrite the expression inside the first pair of parentheses slightly: (4x 2x2) = 2 2 x 2 x x = 2x(2 x)

As (x + 2) = (2 + x), you get

= (4x 2x2) (x + 2) = 2x (2 x)(2 + x) third algebraic identity = 2x (4 x2) = 8x 2x3.

(4x 2x2) (x + 2) = 2x (2 x)(2 + x) third algebraic identity = 2x (4 x2) = 8x 2x3

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The First and Second Algebraic Identity
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Applications of the Algebraic Identities