Sequence and coefficients of a 3-degree polynomial
Source: Vietnam TST 2009, Problem 2
April 21, 2009
algebrapolynomialtrigonometryalgebra proposed
Problem Statement
Let a polynomial P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1 such that the equation P(x) \equal{} 0 has only one real root. A sequence is defined by a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n.
Prove that contains an infinite number of nagetive real numbers.