2004 AMC 12B Problems/Problem 23
Problem
The polynomial has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of
are possible?
Solution 2
Letting the roots be ,
, and
, where
, we see that by Vieta's Formula's,
, and so
. Therefore,
is a factor of
. Letting
gives that
because
. Letting
and noting that
for some
, we see that
is the sum of the roots of
,
and
, and so
. Now, we have that
has roots
and
, and we wish to find the number of possible values of
. By the quadratic formula, we see that
are the two values of noninteger positive real numbers
and
, neither of which is equal to
. This information gives us that
, and so since
is evidently not a square, we have
possible values of
.
Solution 3 (cheese)
Observe that the answer clearly must have something to do with the number , and we see that
is a multiple of
, so there is a very high probability that it is the correct answer.
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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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