Difference between revisions of "2021 AMC 12B Problems/Problem 16"
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==Solution 6== | ==Solution 6== | ||
It is well known that reversing the order of the coefficients of a polynomial turns each root into its corresponding reciprocal. Thus, a polynomial with the desired roots may be written as <math>cx^3+bx^2+a+1</math>. As the problem statement asks for a monic polynomial, our answer is <cmath>\boxed{(\textbf{A}) \frac{1+a+b+c}{c}}</cmath> | It is well known that reversing the order of the coefficients of a polynomial turns each root into its corresponding reciprocal. Thus, a polynomial with the desired roots may be written as <math>cx^3+bx^2+a+1</math>. As the problem statement asks for a monic polynomial, our answer is <cmath>\boxed{(\textbf{A}) \frac{1+a+b+c}{c}}</cmath> | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/M4Ffhp9NLKY?t=923 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==Video Solution by Punxsutawney Phil== | ==Video Solution by Punxsutawney Phil== |
Revision as of 06:50, 4 November 2022
Contents
[hide]Problem
Let be a polynomial with leading coefficient whose three roots are the reciprocals of the three roots of where What is in terms of and
Solution 1
Note that has the same roots as , if it is multiplied by some monomial so that the leading term is they will be equal. We have so we can see that Therefore
Solution 2 (Vieta's bash)
Let the three roots of be , , and . (Here e does NOT mean 2.7182818...) We know that , , and , and that (Vieta's). This is equal to , which equals . -dstanz5
Solution 3 (Fakesolve)
Because the problem doesn't specify what the coefficients of the polynomial actually are, we can just plug in any arbitrary polynomial that satisfies the constraints. Let's take . Then has a triple root of . Then has a triple root of , and it's monic, so . We can see that this is , which is answer choice .
-Darren Yao
Solution 4
If we let and be the roots of , and . The requested value, , is then The numerator is (using the product form of ) and the denominator is , so the answer is
- gting
Solution 5 (Good at Guessing)
The function . If it's , then it becomes So, becomes , so becomes . Also, there is a so the answer must include . The only answer having both of these is .
~smellyman
-Extremelysupercooldude (Minor Latex Edits and Grammar)
Solution 6
It is well known that reversing the order of the coefficients of a polynomial turns each root into its corresponding reciprocal. Thus, a polynomial with the desired roots may be written as . As the problem statement asks for a monic polynomial, our answer is
Video Solution by OmegaLearn
https://youtu.be/M4Ffhp9NLKY?t=923
~ pi_is_3.14
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=vCEJzhDRUoU
Video Solution by OmegaLearn (Vieta's Formula)
Video Solution by Hawk Math
https://www.youtube.com/watch?v=p4iCAZRUESs
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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.