Difference between revisions of "2020 AMC 12B Problems/Problem 17"
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Revision as of 17:54, 14 June 2020
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
FORMAT IT PROPERLY KID
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 12 Problems and Solutions |
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