2021 AMC 12B Problems/Problem 16

Problem 16

Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$

$\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}$

Solution

Note that $f(1/x)$ is very close to $g(x)$ , it just needs to be multiplied by something so that the leading term is $x^3$ . We have $f(1/x) = \frac{1}{x^3} + \frac{a}{x^2}+\frac{b}{x} + c$ so we can see that $g(x) = \frac{x^3}{c}f(1/x)$ Therefore $g(1) = \frac{1}{c}f(1) = \boxed{\textbf{(A) }\frac{1+a+b+c}c}$

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2021 AMC 12B Problems/Problem 16

Problem 16

Solution