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)}\;\dfrac{1 + a + b + c}{c}

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

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

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

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