2008 AIME II Problems/Problem 7
Problem
Let ,
, and
be the three roots of the equation
Find
.
Contents
[hide]Solution
Solution 1
By Vieta's formulas, we have so
Substituting this into our problem statement, our desired quantity is
Also by Vieta's formulas we have
so negating both sides and multiplying through by 3 gives our answer of
Solution 2
By Vieta's formulas, we have , and so the desired answer is
. Additionally, using the factorization
we have that
. By Vieta's again,
Solution 3
Vieta's formulas gives . Since
is a root of the polynomial,
, and the same can be done with
. Therefore, we have
yielding the answer
.
Also, Newton's Sums yields an answer through the application. http://www.artofproblemsolving.com/Wiki/index.php/Newton's_Sums
Solution 4
Expanding, you get:
This looks similar to
Substituting:
Since
,
Substituting, we get
or,
We are trying to find
.
Substituting:
Solution 5
Write and let
. Then
Solving for
and negating the result yields the answer
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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