2021 Fall AMC 12A Problems/Problem 15
Contents
[hide]Problem 15
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
By Vieta's formulas, , and
Since Since
Also, and
Our answer is
~kingofpineapplz
Solution 2
Because all coefficients of are real, , , , and are four zeros of .
First, we compute .
For , following from Vieta's formula,
For , following from Vieta's formula,
Second, we compute .
For , following from Vieta's formula,
For , following from Vieta's formula,
Therefore, .
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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