Difference between revisions of "2003 AIME II Problems/Problem 9"
m (change 6 to 006 since its an aime problem) |
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~ Nafer | ~ Nafer | ||
+ | |||
+ | == Solution 3 == | ||
+ | |||
+ | <math>P(x) = x^{2}Q(x)+x^{4}-x^{3}-x.</math> | ||
+ | |||
+ | So we just have to find: <math>\sum_{n=1}^{4} z^{4}_n - \sum_{n=1}^{4} z^{3}_n - \sum_{n=1}^{4} z_n</math>. | ||
+ | |||
+ | And by [[Newton's Sums]] this computes to: <math>11-4-1 = \boxed{006}</math>. | ||
+ | |||
+ | ~ LuisFonseca123 | ||
== Video Solution by Sal Khan == | == Video Solution by Sal Khan == |
Revision as of 11:47, 28 April 2024
Problem
Consider the polynomials and
Given that
and
are the roots of
find
Solution
When we use long division to divide by
, the remainder is
.
So, since is a root,
.
Now this also follows for all roots of
Now
Now by Vieta's we know that ,
so by Newton's Sums we can find
So finally
Solution 2
Let then by Vieta's Formula we have
By Newton's Sums we have
Applying the formula couples of times yields .
~ Nafer
Solution 3
So we just have to find: .
And by Newton's Sums this computes to: .
~ LuisFonseca123
Video Solution by Sal Khan
https://www.youtube.com/watch?v=ZSESJ8TeGSI&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=14 - AMBRIGGS
[rule]
Nice!-sleepypuppy
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.