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
, 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