1. Consider the polynomial $\displaystyle P(x) = x^5 + ax^3 + bx - c $ where a,b and are real positive numbers

(i) show that P(x) has exactly one real root which is greater than zero

(ii) Hence show that P(x) has four complex roots with at least two of them with negative real parts

2. If the polynomial $\displaystyle x^3 + 3mx^2 + 3nx + r = 0$ has a double root. Show that:

(i) The double root is x = $\displaystyle \frac {mn-r}{2(n-m^2)}$

(ii) $\displaystyle (mn-r)^2 = 4(m^2 -n)(n^2 - mr) $

Thanks