2021 AMC 12A Problems/Problem 12
Contents
Problem
All the roots of the polynomial are positive integers, possibly repeated. What is the value of
?
Solution 1:
By Vieta's formulae, the sum of the 6 roots is 10 and the product of the 6 roots is 16. By inspection, we see the roots are 1, 1, 2, 2, 2, and 2, so the function is . Therefore,
.
~JHawk0224
Solution 2:
Using the same method as solution 1, we find the roots are . Note that
is the negation of the 3rd symmetric sum of the roots; using casework on the number of 1's in each of the
products, we obtain
~ ike.chen.
Video Solution by Hawk Math
https://www.youtube.com/watch?v=AjQARBvdZ20
Video Solution by OmegaLern (Using Vieta's Formulas & Combinatorics)
~ pi_is_3.14
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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.