2023 AMC 10A Problems/Problem 21
Contents
Problem
Let be the unique polynomial of minimal degree with the following properties:
- has a leading coefficient ,
- is a root of ,
- is a root of ,
- is a root of , and
- is a root of .
The roots of are integers, with one exception. The root that is not an integer can be written as , where and are relatively prime integers. What is ?
Solution 1
From the problem statement, we know , and . Therefore, we know that , , and are roots. So, we can factor as , where is the unknown root. Since , we plug in which gives , therefore . Therefore, our answer is
~aiden22gao
~cosinesine
~walmartbrian
~sravya_m18
~ESAOPS
Solution 2
We proceed similarly to solution one. We get that . Expanding, we get that . We know that , so the sum of the coefficients of the cubic expression is equal to one. Thus . Solving for a, we get that a=23/24. Therefore, our answer is
~Aopsthedude
Video Solution 1 by OmegaLearn
Video Solution 2
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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All AMC 10 Problems and Solutions |
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