If (r+1r)^2=3, then r^3+1r^3 equals

1952 AMC 12 Problem 26

Attempt Problem 26 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.

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26.

\left(r+\dfrac1r\right)^2=3,

then

r^3+\dfrac1{r^3}

equals

1

2

0

3

6

Let

u=r+\frac{1}{r}

and express the requested sum in terms of

u

Use

r^3+r^{-3}=u^3-3u

Let

u=r+\frac{1}{r}.

Expanding

u^3

gives

r^3+\frac1{r^3}=u^3-3u.

Since

u^2=3,

this becomes

u(u^2-3)=0.

Thus the value is

0,

so the correct answer is C.