Question

Let $a,$ $b,$ and $c$ be the roots of the polynomial $x^3 + kx + 1.$ What is the sum

$\begin{aligned} &a^3b^2 + a^2b^3 + b^3c^2 \\ &\quad {}+ b^2c^3 + c^3a^2 + c^2a^3? \end{aligned}$

$-k$

$-k + 1$

$1$

$k - 1$

$k$

Answer: E

Concepts:Vieta’s Formulas symmetry (algebra)

Difficulty rating: 2020

Solution:

By Vieta's formulas, $a + b + c = 0,$ $ab + bc + ca = k,$ and $abc = -1.$

Group the sum as $\begin{aligned} &a^2b^2(a + b) + b^2c^2(b + c) \\ &\quad {}+ c^2a^2(c + a). \end{aligned}$ Since $a + b + c = 0,$ we have $a + b = -c,$ $b + c = -a,$ $c + a = -b.$

So the sum equals $\begin{gathered} -a^2b^2 c - ab^2c^2 - a^2bc^2 \\ = -abc(ab + bc + ca) \\ = -(-1)(k) = k. \end{gathered}$

Thus, the correct answer is E.

2025 AMC 12A Problem 19

Answer: E

Difficulty rating: 2020

Solution:

Problem 19 in Other Years