Precalculus · Problem 3.3.3
Polynomial Functions — Stitz & Zeager
Mathematics — Precalculus · real-zeros-polynomials
Problem
For the given polynomial: • Use Cauchy's Bound to find an interval containing all of the real zeros. • Use the Rational Zeros Theorem to make a list of possible rational zeros. • Use Descartes' Rule of Signs to list the possible number of positive and negative real zeros, counting multiplicities.Intuitive explanation
Cauchy's Bound gives you a simple way to fence in where all real zeros must live by looking at the polynomial's coefficients. The Rational Zeros Theorem narrows down which rational candidates are even worth testing—they must be ratios of factors of the constant term and leading coefficient. Finally, Descartes' Rule counts sign changes in and to predict how many positive and negative zeros could actually exist, eliminating impossible scenarios before you do any real work.
Intuitive explanation
Cauchy's Bound gives you a simple way to fence in where all real zeros must live by looking at the polynomial's coefficients. The Rational Zeros Theorem narrows down which rational candidates are even worth testing—they must be ratios of factors of the constant term and leading coefficient. Finally, Descartes' Rule counts sign changes in and to predict how many positive and negative zeros could actually exist, eliminating impossible scenarios before you do any real work.