2023 AMC 10A Problems/Problem 21
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 
?
[bold] Solution 1 [\bold]
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
