2021 Fall AMC 12A Problems/Problem 15
Contents
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 1
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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