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smc

algebra intermediate

Problem

The expression

a^{3} - a^{- 3}

equals:

(A)

(a - \frac{1}{a}) (a^{2} + 1 + \frac{1}{a ^{2}})

(B)

(\frac{1}{a} - a) (a^{2} - 1 + \frac{1}{a ^{2}})

(C)

(a - \frac{1}{a}) (a^{2} - 2 + \frac{1}{a ^{2}})

(D)

(\frac{1}{a} - a) (\frac{1}{a ^{2}} + 1 + a^{2})

Solution

Recall that

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

. Letting

a = x

and

a^{- 1} = y

, we find that

a^{3} - a^{- 3} = (a - \frac{1}{a}) (a^{2} + 1 + \frac{1}{a ^{2}})

.

Final answer

A