Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.
Source: 2014 Saudi Arabia Pre-TST 2.2
Tags: algebra, polynomial
Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.