2021 AMC 12B Problem 16

Problem 16 of 25IntermediateAlgebra

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\lt a\lt b\lt c.

What is

g(1)

in terms of

a,

b,

and

c?

Show solution

Solution

Let

f

have roots

r,s,t.

Since

g

is monic with roots

\tfrac1r,\tfrac1s,\tfrac1t,

\begin{aligned} g(1) &= \left(1-\tfrac1r\right)\left(1-\tfrac1s\right) \\ &\quad {}\cdot \left(1-\tfrac1t\right) \\ &= \dfrac{(r-1)(s-1)(t-1)}{rst}. \end{aligned}

Now

f(1)=(1-r)(1-s)(1-t)

=1+a+b+c,

(r-1)(s-1)(t-1)

=-(1+a+b+c).

Also

rst=-c.

Therefore

g(1)

=\dfrac{-(1+a+b+c)}{-c}

=\dfrac{1+a+b+c}{c}.

Thus, the correct answer is A.

Tagged: Vieta’s Formulas · polynomial