2017 AMC 10A Problems/Problem 24

Revision as of 17:32, 8 February 2017 by Anematode (talk | contribs) (Solution)

Problem

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$?

$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

Solution

$f(x)$ must have four roots, three of which are roots of $g(x)$. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of $f(x)$ and $g(x)$ are the same, we know that

\[f(x)=g(x)(x-r)\]

where $r\in\mathbb{C}$ is the fourth root of $f(x)$. Substituting $g(x)$ and expanding, we find that

\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*}

Comparing coefficients with $f(x)$, we see that

\begin{align*} a-r&=1\\ 1-ar&=b\\ 10-r&=100\\ -10r&=c\\ \end{align*}