1975 AMC 12 Problem 27

Attempt Problem 27 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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27.

If

p,

q,

and

r

are distinct roots of

x^3-x^2+x-2=0,

then

p^3+q^3+r^3

equals

-1

1

3

5

none of these

Answer: E

Concepts:polynomial Vieta’s Formulas Newton’s Sums

Difficulty rating: 1960

Small hint:

Use Vieta to find

p+q+r

and

pq+pr+qr

Big hint:

First compute

p^2+q^2+r^2,

then add the three equations satisfied by the roots

Solution:

Vieta gives

p+q+r=1

and

pq+pr+qr=1.

Thus

\begin{aligned} p^2+q^2+r^2 &=(p+q+r)^2\\ &\quad-2(pq+pr+qr)\\ &=-1. \end{aligned}

Each root

t

satisfies

t^3=t^2-t+2.

Summing over the three roots yields

p^3+q^3+r^3=-1-1+6=4.

This is not among choices A–D.

Therefore, the correct answer is E.