2007 AIME I Problems/Problem 8
Problem
The polynomial is cubic. What is the largest value of for which the polynomials and are both factors of ?
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 .
Solution 3
Since and are both factors of , which is cubic, we know the other factors associated with each of and must be linear. Let , where and . Then we have that . Equating coefficients, we get the following system of equations:
Using equations and to make substitutions into equation , we see that the 's drop out and we're left with . Substituting this expression for into equation and solving, we see that must be .
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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 |
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