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 +...") |
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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 + s^2 + t^2)(rst)\right|.</cmath> | 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 + s^2 + t^2)(rst)\right|.</cmath> | ||
| − | ==Solution== | + | ==Solution 1== |
| + | |||
| + | From Vieta's formulas, we have <math>r+s+t = 15, rs+rt+st = 4, </math> and <math>rst = -4.</math> Now, note that | ||
| + | <cmath>\begin{align*} | ||
| + | \left|\left(r^2+s^2+t^2\right)(rst)\right| &= \left|\left(\left(r+s+t\right)^2-2(rs+rt+st)\right)\right|\\ | ||
| + | &=\left((15^2-2\cdot4\right)(-4)\\ | ||
| + | &=\left|217\cdot(-4)\right| = \boxed{868}.\end{align*}</cmath> | ||
| + | |||
| + | ~SMO_Team | ||
| + | |||
| + | ==Solution 2== | ||
| + | By Vieta'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>. | ||
| + | |||
| + | -Vivdax | ||
Latest revision as of 14:26, 10 September 2025
Problem
The polynomial 
has distinct real roots 
, 
, and 
. Find the value of ![\[\left|(r^2 + s^2 + t^2)(rst)\right|.\]](http://latex.artofproblemsolving.com/0/5/5/055d7c5b74c781f9c0247eacb7b3412345ae4a4b.png)
Solution 1
From Vieta's formulas, we have 
and 
Now, note that
~SMO_Team
Solution 2
By Vieta's, we have that 
and 
. Thus, 
.
-Vivdax
