Difference between revisions of "2007 AIME I Problems/Problem 8"
| Line 9: | Line 9: | ||
We then know that <math>a</math> is a root of | We then know that <math>a</math> is a root of | ||
| − | + | <math> | |
Q_{2}(x)-2Q_{1}(x) = 2x^{2}+2kx-43x+k-2x^{2}-2kx+58x+2k = 15x+3k = 0 | Q_{2}(x)-2Q_{1}(x) = 2x^{2}+2kx-43x+k-2x^{2}-2kx+58x+2k = 15x+3k = 0 | ||
| − | + | </math> | |
, so <math>x = \frac{-k}{5}</math>. | , so <math>x = \frac{-k}{5}</math>. | ||
We then know that <math>\frac{-k}{5}</math> is a root of <math>Q_{1}</math> so we get: | We then know that <math>\frac{-k}{5}</math> is a root of <math>Q_{1}</math> so we get: | ||
| − | + | <math> | |
\frac{k^{2}}{25}+(k-29)\left(\frac{-k}{5}\right)-k = 0 = k^{2}-5(k-29)(k)-25k = k^{2}-5k^{2}+145k-25k | \frac{k^{2}}{25}+(k-29)\left(\frac{-k}{5}\right)-k = 0 = k^{2}-5(k-29)(k)-25k = k^{2}-5k^{2}+145k-25k | ||
| − | + | </math> | |
or <math>k^{2}=30k</math>, so <math>k=30</math> is the highest. | or <math>k^{2}=30k</math>, so <math>k=30</math> is the highest. | ||
| − | We can trivially check into the original equations to find that <math>k=30</math> produces a root in common, so the answer is <math> | + | We can trivially check into the original equations to find that <math>k=30</math> produces a root in common, so the answer is <math>030</math>. |
== See also == | == See also == | ||
{{AIME box|year=2007|n=I|num-b=7|num-a=9}} | {{AIME box|year=2007|n=I|num-b=7|num-a=9}} | ||
Revision as of 08:26, 15 March 2007
Problem
The polynomial 
is cubic. What is the largest value of 
for which the polynomials 
and 
are both factors of ?
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it. We can see that they must have a root in common for them to both be factors of the same cubic.
Let this root be 
.
We then know that is a root of
, so 
.
We then know that 
is a root of 
so we get:
or 
, so 
is the highest.
We can trivially check into the original equations to find that produces a root in common, so the answer is 
.
See also
| 2007 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
