2021 Fall AMC 12B Problems/Problem 14
Contents
Problem
Suppose that , and are polynomials with real coefficients, having degrees , , and , respectively, and constant terms , , and , respectively. Let be the number of distinct complex numbers that satisfy the equation . What is the minimum possible value of ?
Solution
The answer cannot be as every nonconstant polynomial has at least distinct complex root (Fundamental Theorem of Algebra). Since has degree we conclude that has degree and is thus nonconstant.
It now suffices to illustrate an example for which : Take Note that has degree and constant term so it satisfies the conditions.
We need to find the solutions to Clearly, the only distinct complex root is so our answer is
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Video Solution
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See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |
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