2021 Fall AMC 12A Problems/Problem 15
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[hide]Problem
Recall that the conjugate of the complex number , where and are real numbers and , is the complex number . For any complex number , let . The polynomial has four complex roots: , , , and . Let be the polynomial whose roots are , , , and , where the coefficients and are complex numbers. What is
Solution 1
By Vieta's formulas, , and
Since
By Vieta's formulas, , and
Since Since
Our answer is
~kingofpineapplz ~sl_hc
Solution 2
Since the coefficients of are real, the roots of can also be written as . With this observation, it's easy to see that the polynomials and have the same roots. Hence, there exists some constant such that
By comparing coefficients, its easy to see that . Hence and . Hence , , so and our answer is .
~tsun26
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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All AMC 12 Problems and Solutions |
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