Difference between revisions of "2010 AMC 12A Problems/Problem 21"
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Revision as of 21:48, 3 July 2013
Problem
The graph of 
lies above the line 
except at three values of 
, where the graph and the line intersect. What is the largest of these values?
Solution
The values in which intersect at are the same as the zeros of 
.
Since there are 
zeros and the function is never negative, all zeros must be double roots because the function's degree is 
.
Suppose we let 
, 
, and 
be the roots of this function, and let 
be the cubic polynomial with roots , , and .
In order to find 
we must first expand out the terms of 
.
![\[(x^3-ux^2+vx-w)^2\]](http://latex.artofproblemsolving.com/d/1/6/d16c66369fc2eddcf00659452090770548ed7b2f.png)
[Quick note: Since we don't know 
, 
, and 
, we really don't even need the last 3 terms of the expansion.]
All that's left is to find the largest root of 
.
See also
| 2010 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
