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Revision as of 17:54, 12 November 2023
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
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 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.