2024 AMC 12A Problems/Problem 15
Contents
[hide]Problem
The roots of are and What is the value of
Solution 1
You can factor as .
For any polynomial , you can create a new polynomial , which will have roots that instead have the value subtracted.
Substituting and into for the first polynomial, gives you and as for both equations. Multiplying and together gives you .
-ev2028
~Latex by eevee9406
Solution 2
Let . Then .
We find that and , so .
Select D
~eevee9406
Solution 3
First, denote that Then we expand the expression ~lptoggled
Solution 4 (Reduction of power)
The motivation for this solution is the observation that is easy to compute for any constant c, since (*), where is the polynomial given in the problem. The idea is to transform the expression involving into one involving .
Since is a root of , which gives us that . Then Since and are also roots of , the same analysis holds, so
(*) This is because since for all .
~tsun26 ~KSH31415 (final step and clarification)
Solution 5 (Cheesing it out)
Expanding the expression gives us
Notice that everything other than is a multiple of . Solving for using vieta's formulas, we get . Since is , the answer should be as well. The only answer that is is .
~callyaops
Solution 6
Suppose
then . Substitute into (It is same for because the squares in )
whose constant is 125
according to Vieta's theorem,
.
~JiYang
See also
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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All AMC 12 Problems and Solutions |
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