2017 AMC 10A Problems/Problem 24
Problem
For certain real numbers ,
, and
, the polynomial
has three distinct roots, and each root of
is also a root of the polynomial
What is
?
Solution
must have four roots, three of which are roots of
. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of
and
are the same, we know that
where is the fourth root of
. Substituting
and expanding, we find that
Comparing coefficients with , we see that
Let's solve for and
. Since
,
, so
. Since
,
, and
. Thus, we know that
Taking , we find that
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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All AMC 10 Problems and Solutions |
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