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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 $(a + b) (a - b)$ ?

\begin{array}{l} (a + b) (a - b) & = a^{2} - a b + b a - b^{2} \\ = a^{2} - b^{2} \end{array}

Rule

The Third Algebraic Identity

(a + b) (a - b) = a^{2} - b^{2}

Example 1

Expand $(\sqrt{x} + 1) (\sqrt{x} - 1)$

If you use the third algebraic identity, you get

\begin{array}{l} (\sqrt{x} + 1) (\sqrt{x} - 1) & = {(\sqrt{x})}^{2} - 1^{2} \\ = x - 1 \end{array}

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

Example 2

Expand $(4 x - 2 x^{2}) (x + 2)$

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:

\begin{array}{l} (4 x - 2 x^{2}) & = 2 \cdot 2 \cdot x - 2 \cdot x \cdot x \\ = 2 x (2 - x) \end{array}

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

\begin{array}{l} (4 x - 2 x^{2}) (x + 2) \\ = 2 x \cdot \underset{third algebraic identity}{\underset{⏟}{(2 - x) (2 + x)}} \\ = 2 x (4 - x^{2}) \\ = 8 x - 2 x^{3} . \end{array}

\begin{array}{l} (4 x - 2 x^{2}) (x + 2) & = 2 x \cdot \underset{third algebraic identity}{\underset{⏟}{(2 - x) (2 + x)}} \\ = 2 x (4 - x^{2}) \\ = 8 x - 2 x^{3} \end{array}

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The First and Second Algebraic Identity

Applications of the Algebraic Identities