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Difference between revisions of "2024 SSMO Speed Round Problems/Problem 3"

(Created page with "==Problem== The polynomial <math>x^3 - 15x^2 + 4x + 4</math> has distinct real roots <math>r</math>, <math>s</math>, and <math>t</math>. Find the value of <cmath>\left|(r^2 +...")
 
(→‎Solution)
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==Solution==
 
==Solution==
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By vietta's, we have that <math>r^2+s^2+t^2 = (r+s+t)^2 - 2(rs+st+rt) = 15^2 - 2(4) = 217</math> and <math>rst = -4</math>. Thus, <math>|(r^2+s^2+t^2)(rst)| = |217\cdot -4| = \boxed{868}</math>.

Revision as of 19:34, 2 May 2025

Problem

The polynomial $x^3 - 15x^2 + 4x + 4$ has distinct real roots $r$, $s$, and $t$. Find the value of \[\left|(r^2 + s^2 + t^2)(rst)\right|.\]

Solution

By vietta's, we have that $r^2+s^2+t^2 = (r+s+t)^2 - 2(rs+st+rt) = 15^2 - 2(4) = 217$ and $rst = -4$. Thus, $|(r^2+s^2+t^2)(rst)| = |217\cdot -4| = \boxed{868}$.

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