Question

State the number of complex roots,possible number of real roots and possible rational roots for x^5 - x^3 - 11x^2+ 9x+18=0

Answer 1

there are 5 possible rational and real roots. Given the wording of the problem, though, it's likely that there are some complex roots, reducing the number of real candidates.

There may be 2 or 4 complex roots, meaning that there may be 5,3, or 1 real root.

Lacking any further information, all the real roots may be rational, and in fact will be integers.

A little synthetic division shows that we have no rational roots.

In fact, there are no real roots greater than 0 or less than -1. So, there are 4 complex roots and one real root -1 < r < 0