Precalculus · Problem 3.3.7
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, the Rational Zeros Theorem, and Descartes' Rule of Signs are three complementary tools that narrow down where zeros could be before you start testing. Cauchy's Bound gives you a safe interval to search in; the Rational Zeros Theorem lists only the candidates worth testing (using the ratio of factors of the constant term to factors of the leading coefficient); and Descartes' Rule counts sign changes in and to predict how many positive and negative zeros are actually possible. Together, they transform a blind search into a strategic one.
Intuitive explanation
Cauchy's Bound, the Rational Zeros Theorem, and Descartes' Rule of Signs are three complementary tools that narrow down where zeros could be before you start testing. Cauchy's Bound gives you a safe interval to search in; the Rational Zeros Theorem lists only the candidates worth testing (using the ratio of factors of the constant term to factors of the leading coefficient); and Descartes' Rule counts sign changes in and to predict how many positive and negative zeros are actually possible. Together, they transform a blind search into a strategic one.