# Difference between revisions of "1975 AHSME Problems/Problem 27"

Henry wang (talk | contribs) (Created page with "If <math>p</math> is a root of <math>x^3 - x^2 + x - 2 = 0</math>, then <math>p^3 - p^2 + p - 2 = 0</math>, or <cmath>p^3 = p^2 - p + 2.</cmath> Similarly, <math>q^3 = q^2 - q...") |
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+ | ==Problem== | ||

+ | If <math>p, q</math> and <math>r</math> are distinct roots of <math>x^3-x^2+x-2=0</math>, then <math>p^3+q^3+r^3</math> equals | ||

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+ | <math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}</math> | ||

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If <math>p</math> is a root of <math>x^3 - x^2 + x - 2 = 0</math>, then <math>p^3 - p^2 + p - 2 = 0</math>, or | If <math>p</math> is a root of <math>x^3 - x^2 + x - 2 = 0</math>, then <math>p^3 - p^2 + p - 2 = 0</math>, or | ||

<cmath>p^3 = p^2 - p + 2.</cmath> | <cmath>p^3 = p^2 - p + 2.</cmath> |

## Revision as of 21:52, 12 February 2021

## Problem

If and are distinct roots of , then equals

If is a root of , then , or
Similarly, , and , so

By Vieta's formulas, , , and . Squaring the equation , we get Subtracting , we get

Therefore, . The answer is (E).