MathDB

Polynomials with common rootas [Spain 2000]

Source:

February 3, 2011

algebrapolynomialparameterizationquadraticsalgebra unsolved

Problem Statement

Consider the polynomials P(x) = x^4 + ax^3 + bx^2 + cx + 1 \text{and} Q(x) = x^4 + cx^3 + bx^2 + ax + 1. Find the conditions on the parameters

a, b,

c with

a\neq c

for which

P(x)

and

Q(x)

have two common roots and, in such cases, solve the equations

P(x) = 0

and

Q(x) = 0.