Difference between revisions of "1982 AHSME Problems/Problem 27"
MRENTHUSIASM (talk | contribs) (Created page with "== Problem == Suppose <math>z=a+bi</math> is a solution of the polynomial equation <cmath>c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0,</cmath> where <math>c_0, c_1, c_2, c_3, a,</math>...") |
MRENTHUSIASM (talk | contribs) |
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== Solution == | == Solution == | ||
+ | Let <math>z=wi,</math> so the given polynomial equation becomes | ||
+ | <cmath>\begin{align*} | ||
+ | c_4(wi)^4+ic_3(wi)^3+c_2(wi)^2+ic_1(wi)+c_0&=0 \\ | ||
+ | c_4w^4+c_3w^3-c_2w^2-c_1w+c_0&=0, &&(\bigstar) | ||
+ | \end{align*}</cmath> | ||
+ | which is a polynomial equation in <math>w</math> with real coefficients. | ||
+ | |||
+ | We are given that <math>w=\frac{a+bi}{i}=b-ai</math> is a solution to <math>(\bigstar).</math> By the Complex Conjugate Root Theorem, we conclude that <math>w=b+ai</math> must also be a solution to <math>(\bigstar),</math> from which <math>z=(b+ai)i=\boxed{\textbf{(C)}\ -a+bi}</math> must also be a solution to the given polynomial equation. | ||
+ | |||
+ | ~MRENTHUSIASM | ||
== See Also == | == See Also == | ||
{{AHSME box|year=1982|num-b=26|num-a=28}} | {{AHSME box|year=1982|num-b=26|num-a=28}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 02:26, 7 September 2021
Problem
Suppose is a solution of the polynomial equation where and are real constants and Which of the following must also be a solution?
Solution
Let so the given polynomial equation becomes which is a polynomial equation in with real coefficients.
We are given that is a solution to By the Complex Conjugate Root Theorem, we conclude that must also be a solution to from which must also be a solution to the given polynomial equation.
~MRENTHUSIASM
See Also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.