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Gunga P15

Source:

October 16, 2021

MOAA 2021Gunga

Problem Statement

Let

a,b,c,d

be the four roots of the polynomial

x^4+3x^3-x^2+x-2.

Given that

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}

and

\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}

, the value of

\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}

can be expressed as

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Compute

m+n

.
Proposed by Nathan Xiong

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