1971 AMC 12 Problem 22

Attempt Problem 22 of the 1971 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 1971 AMC 12 solutions, or check the answer key.

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22.

w

is one of the imaginary roots of

x^3=1,

then

(1-w+w^2)(1+w-w^2)

is equal to:

4

w

2

w^2

1

Use

w^2+w+1=0

Replace

1+w^2

-w

and

1+w

-w^2

Because

w\ne1

and

w^3=1,

we have

1+w+w^2=0.

Thus

\begin{gathered} 1-w+w^2=-2w,\\ 1+w-w^2=-2w^2. \end{gathered}

Their product is

4w^3=4.

Therefore, the correct answer is A.

Problem 22 in other years