1977 USAMO Problems/Problem 3
If and are two of the roots of , prove that is a root of .
Given the roots of the equation .
First, Vieta's relations give .
Then and .
The other coefficients give or .
Let and , so (1).
Second, is a root, and is a root, .
Multiplying: or .
In (1): .
Conclusion: is a root of .
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