Difference between revisions of "2023 AMC 10A Problems/Problem 21"
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Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties: | Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties: | ||
Revision as of 21:44, 9 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 
,
is a root of 
, and
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. Because of this, we can factor as 
, where 
is the unknown root. Plugging in 
gives 
, so 
. Therefore, our answer is 
, or 
~aiden22gao
~cosinesine
~walmartbrian
Video Solution 1 by OmegaLearn
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. 
