Difference between revisions of "2017 AMC 10A Problems/Problem 24"
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<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | a-r=1\ | + | a-r&=1\ |
− | 1-ar=b\ | + | 1-ar&=b\ |
− | 10-r=100\ | + | 10-r&=100\ |
− | -10r=c\ | + | -10r&=c\ |
\end{align*}</cmath> | \end{align*}</cmath> |
Revision as of 16:31, 8 February 2017
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