2021 Fall AMC 12B Problems/Problem 14
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); the polynomial has degree , so the polynomial has degree and is thus nonconstant. It now suffices to illustrate an example for which . Take and
has degree 6 and constant term , so it satisfies the conditions. We need to find the solutions to or Clearly, there is one distinct complex root, , so our answer is
~kingofpineapplz and kgator
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.