2019 AMC 12A Problems/Problem 17
Problem
Let denote the sum of the th powers of the roots of the polynomial . In particular, , , and . Let , , and be real numbers such that for , , What is ?
Solution 1
Applying Newton's Sums (see this link), we havesowe get the answer as .
Solution 2
Let , and be the roots of the polynomial. Then,
Adding these three equations, we get
can be written as , giving
We are given that is satisfied for , , , meaning it must be satisfied when , giving us .
Therefore, , and by matching coefficients.
.
Solution 3
Let , and be the roots of the polynomial. By Vieta's Formulae, we have
.
We know . Consider .
Using and , we see .
We have
Rearrange to get
So, .
-gregwwl
Solution 4
Let be the roots of . Then:
\\ \\
If we multiply both sides of the equation by , where is a positive integer, then that won't change the coefficients, but just the degree of the new polynomial and the other term's exponents. We can try multiplying to find , but that is just to check. So then with the above information about , we see that:
, ,
Then:
This means that , as expected. So we have . So our answer is
-IzhanAli
Video Solution
For those who want a video solution: https://www.youtube.com/watch?v=tAS_DbKmtzI
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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