2020 AMC 12B Problems/Problem 17
Problem
How many polynomials of the form , where
,
,
, and
are real numbers, have the property that whenever
is a root, so is
? (Note that
)
Solution
Let . We first notice that
, so in order
to be a root of
,
must also be a root of P, meaning that 3 of the roots of
must be
,
,
. However, since
is degree 5, there must be two additional roots. Let one of these roots be
, if
is a root, then
and
must also be roots. However,
is a fifth degree polynomial, and can therefore only have
roots. This implies that
is either
,
, or
. Thus we know that the polynomial
can be written in the form
. Moreover, by Vieta's, we know that there is only one possible value for the magnitude of
as
, meaning that the amount of possible polynomials
is equivalent to the possible sets
. In order for the coefficients of the polynomial to all be real,
due to
and
being conjugates and since
, (as the polynomial is 5th degree) we have two possible solutions for
which are
and
yielding two possible polynomials. The answer is thus
.
~Murtagh