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

(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==
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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 22:52, 12 February 2021

Problem

If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals

$\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}$



If $p$ is a root of $x^3 - x^2 + x - 2 = 0$, then $p^3 - p^2 + p - 2 = 0$, or \[p^3 = p^2 - p + 2.\] Similarly, $q^3 = q^2 - q + 2$, and $r^3 = r^2 - r + 2$, so \[p^3 + q^3 + r^3 = (p^2 + q^2 + r^2) - (p + q + r) + 6.\]

By Vieta's formulas, $p + q + r = 1$, $pq + pr + qr = 1$, and $pqr = 2$. Squaring the equation $p + q + r = 1$, we get \[p^2 + q^2 + r^2 + 2pq + 2pr + 2qr = 1.\] Subtracting $2pq + 2pr + 2qr = 2$, we get \[p^2 + q^2 + r^2 = -1.\]

Therefore, $p^3 + q^3 + r^3 = (p^2 + q^2 + r^2) - (p + q + r) + 6 = (-1) - 1  + 6 = \boxed{4}$. The answer is (E).

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