Difference between revisions of "2021 Fall AMC 12A Problems/Problem 15"
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− | For | + | ==Problem== |
− | <cmath>z^3 + | + | Recall that the conjugate of the complex number <math>w = a + bi</math>, where <math>a</math> and <math>b</math> are real numbers and <math>i = \sqrt{-1}</math>, is the complex number <math>\overline{w} = a - bi</math>. For any complex number <math>z</math>, let <math>f(z) = 4i\hspace{1pt}\overline{z}</math>. The polynomial <cmath>P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1</cmath> has four complex roots: <math>z_1</math>, <math>z_2</math>, <math>z_3</math>, and <math>z_4</math>. Let <cmath>Q(z) = z^4 + Az^3 + Bz^2 + Cz + D</cmath> be the polynomial whose roots are <math>f(z_1)</math>, <math>f(z_2)</math>, <math>f(z_3)</math>, and <math>f(z_4)</math>, where the coefficients <math>A,</math> <math>B,</math> <math>C,</math> and <math>D</math> are complex numbers. What is <math>B + D?</math> |
+ | |||
+ | <math>(\textbf{A})\: {-}304\qquad(\textbf{B}) \: {-}208\qquad(\textbf{C}) \: 12i\qquad(\textbf{D}) \: 208\qquad(\textbf{E}) \: 304</math> | ||
==Solution== | ==Solution== |
Latest revision as of 19:01, 24 May 2023
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
By Vieta's formulas, , and
Since
Since
Also, and
Our answer is
~kingofpineapplz
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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.