1952 AMC 12 Problem 3

Attempt Problem 3 of the 1952 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1952 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The expression

a^3-a^{-3}

equals:

\left(a-\dfrac1a\right)\left(a^2+1+\dfrac1{a^2}\right)

\left(\dfrac1a-a\right)\left(a^2-1+\dfrac1{a^2}\right)

\left(a-\dfrac1a\right)\left(a^2-2+\dfrac1{a^2}\right)

\left(\dfrac1a-a\right)\left(\dfrac1{a^2}+1+a^2\right)

None of these

View the expression as a difference of two cubes

Use

u^3-v^3=(u-v)(u^2+uv+v^2)

with

v=\frac{1}{a}

Apply the difference-of-cubes identity with

u=a

and

v=\frac{1}{a}.

Since

uv=1,

\begin{aligned} a^3-\frac1{a^3} &=\left(a-\frac1a\right)\\ &\quad{}\cdot\left(a^2+1+\frac1{a^2}\right). \end{aligned}

Thus, the correct answer is A.

Problem 3 in other years