1. Consider the polynomial 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 has a double root. Show that:
(i) The double root is x =
(ii)
Thanks
Note that the same works for the negative numbers since is symmetric about the y-axis. Namely, . Appealing to Rolle's theorem again guarantees that . It follows that . But, by the FTA we must have that has five zeros counting multiplicity. But, by a very easy to prove theorem, we have that since the coefficients of are real that the zeros come in conjugate pairs, so that . The negative part buisness follows since since they aren't real. Thus, either or . A similar analysis shows that either has negative imaginary parts.