Difference between revisions of "2017 AMC 10A Problems/Problem 24"
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where <math>r\in\mathbb{C}</math> is the fourth root of <math>f(x)</math>. Substituting <math>g(x)</math> and expanding, we find that | where <math>r\in\mathbb{C}</math> is the fourth root of <math>f(x)</math>. Substituting <math>g(x)</math> and expanding, we find that | ||
− | <cmath>f(x)=(x^3+ax^2+x+10)(x-r)=x^4+(a-r)x^3+(1-ar)x^2+(10-r)x-10r.</cmath> | + | <cmath>\begin{align*}f(x)&=(x^3+ax^2+x+10)(x-r)\ |
+ | &=x^4+(a-r)x^3+(1-ar)x^2+(10-r)x-10r.\end{align*}</cmath> | ||
Comparing coefficients with <math>f(x)</math>, we see that | Comparing coefficients with <math>f(x)</math>, we see that |
Revision as of 16:32, 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