Difference between revisions of "2020 AMC 12B Problems/Problem 17"
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Revision as of 18:54, 14 June 2020
Problem
How many polynomials of the form , where , , , and are real numbers, have the property that whenever is a root, so is ? (Note that )
Solution
Let . We first notice that , so in order to be a root of , must also be a root of P, meaning that 3 of the roots of must be , , . However, since is degree 5, there must be two additional roots. Let one of these roots be , if is a root, then and must also be roots. However, is a fifth degree polynomial, and can therefore only have roots. This implies that is either , , or . Thus we know that the polynomial can be written in the form . Moreover, by Vieta's, we know that there is only one possible value for the magnitude of as , meaning that the amount of possible polynomials is equivalent to the possible sets . In order for the coefficients of the polynomial to all be real, due to and being conjugates and since , (as the polynomial is 5th degree) we have two possible solutions for which are and yielding two possible polynomials. The answer is thus .
~Murtagh
FORMAT IT PROPERLY KID
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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