AMC 12B 2021 Spring (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

\text{(A)}\;\frac{1+a+b+c}{c}

\text{(B)}\;\text{1+a+b+c}

\text{(C)}\;\frac{1+a+b+c}{c^2}

\text{(D)}\;\frac{a+b+c}{c^2}

\text{(E)}\;\frac{1+a+b+c}{a+b+c}