Question

Find the roots of the polynomial eq. x^3-2x^2+10x+136=0

Answer 1

First, prove that it has any Real solutions. Decide if there MUST BE a positive of negative solution.

Second, we like it when there is a Rational solution, so look for those, first. You possible rational solutions come from all possible factors of 136.

Third, well, let's get this far before we decide what's next. It may be trivial.

Answer 2

By the Descartes rule of signs, we know that there are either two positive roots, or two complex roots, and there is definitely one negative root, i.e. a factor in the form (x-a).

By the rational root theorem, you will need to try to find a root which is the product of one or more factors of the constant term 136, i.e. 1,2,4,8,17.

Once you find a root, you can use synthetic division to reduce the equation to a quadratic, which can then be readily solved.