Question

Using complete sentences, explain how the Rational Root Theorem and Descartes’ Rule of Signs are used to find the zeros of a polynomial function. (2 points)

Answer 1

I was typing up a proof for this, then realized we only wish to know how to find these. Using the rational root theorem, we can find the possible rational roots of a polynomial, and using Descartes' Rule of Signs allows us to find the total amount of positive and negative solutions to such a polynomial. Using information from both of these, we apply Synthetic Division (Ruffini's Rule) or Long division to reduce the polynomial, further, thus allowing us to simplify continually. Since, by the well-ordering principle, this process must end, we can then solve a polynomial with two irrational or any set of imaginary roots, by the same process.