1977 AHSME Problems/Problem 23

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Solution

Let $r_1$ and $r_2$ be the roots of the equation $x^2+mx+n=0$. Then, $r_1^3$ and $r_2^3$ are the roots of $x^2+px+q=0$. Applying Vieta's Formulas to the first equation, we have $r_1 + r_2 = -m$ and $r_1r_2 = n$. Similarly, from the second equation, we have $r_1^3 + r_2^3 = -p$ and $r_1^3r_2^3 = q$. Cubing $r_1 + r_2 = -m$, we get $r_1^3 + r_2^3 + 3r_1r_2(r_1+r_2) = -m^3$. Plugging in from the other equations gives $-p + 3n(-m)=-m^3$. Rearranging we have $\boxed{\textbf{(B)} p=m^3-3mn.}$

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