2007 AIME I Problems/Problem 8
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Problem
The polynomial 
is cubic. What is the largest value of 
for which the polynomials 
and 
are both factors of ?
Contents |
Solution
Solution 1
We can see that 
and 
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 
.
Solution 2
Again, let the common root be ; let the other two roots be 
and 
. We can write that 
and that 
.
Therefore, we can write four equations (and we have four variables), 
, 
, 
, and 
.
The first two equations show that 
. The last two equations show that 
. Solving these show that 
and that 
. Substituting back into the equations, we eventually find that 
.
See also
| 2007 AIME I (Problems • Resources) | ||
| Preceded by Problem 7 | Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
