Question

Suppose that the roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are $\cos\tfrac{2\pi}{7},$ $\cos\tfrac{4\pi}{7},$ and $\cos\tfrac{6\pi}{7},$ where angles are in radians. What is $abc?$

$-\dfrac{3}{49}$

$-\dfrac{1}{28}$

$\dfrac{\sqrt[3]{7}}{64}$

$\dfrac{1}{32}$

$\dfrac{1}{28}$

Answer: D

Concepts:roots of unity polynomial Vieta’s Formulas

Difficulty rating: 2450

Solution:

Let the three roots be $r_1,r_2,r_3.$ The seventh roots of unity give $r_1+r_2+r_3=-\tfrac12.$ Doubling the three angles merely permutes their cosines, so $\sum r_i^2=\tfrac32+\tfrac12\sum r_i=\tfrac54.$ Hence $r_1r_2+r_1r_3+r_2r_3$ equals $\tfrac12((\sum r_i)^2-\sum r_i^2)=-\tfrac12.$

The product-to-sum identity gives $4r_1r_2r_3=1+r_1+r_2+r_3=\tfrac12,$ so $r_1r_2r_3=\tfrac18.$ Therefore their monic polynomial is $x^3 + \tfrac12 x^2 - \tfrac12 x - \tfrac18 = 0.$

Matching coefficients, $a = \tfrac12,$ $b = -\tfrac12,$ $c = -\tfrac18.$ Therefore $abc = \tfrac12\cdot\left(-\tfrac12\right)\cdot\left(-\tfrac18\right) = \tfrac{1}{32}.$

Thus, the correct answer is D.

2021 AMC 12A Spring Problem 22

Answer: D

Difficulty rating: 2450

Solution:

Problem 22 in Other Years