Post on 17-Jan-2016
Copyright © 2011 Pearson Education, Inc.
The Theoryof Equations
Section 3.3
Polynomial and Rational Functions
Copyright © 2011 Pearson Education, Inc. Slide 3-3
3.3
Definition: MultiplicityIf the factor x – c occurs k times in the complete
factorization of the polynomial P(x), then c is called a
root of P(x) = 0 with multiplicity k.
n-Root TheoremIf P(x) = 0 is a polynomial equation with real or complex
coefficients and positive degree n, then, when multiplicity
is considered, P(x) = 0 has n roots.
The Number of Rootsof a Polynomial Equation
Copyright © 2011 Pearson Education, Inc. Slide 3-4
3.3
Conjugate Pairs TheoremIf P(x) = 0 is a polynomial equation with real coefficients
and the complex number a + bi (b ≠ 0) is a root, then a – bi
is also a root.
The Conjugate Pairs Theorem
Copyright © 2011 Pearson Education, Inc. Slide 3-5
3.3
Descartes’s Rule of SignsSuppose P(x) = 0 is a polynomial equation with real
coefficients and with terms written in descending order. The number of positive real roots of the equation is either
equal to the number of variations of sign of P(x) or less than that by an even number.
The number of negative real roots of the equation is either equal to the number of variations of sign of P(–x) or less than that by an even number.
Descartes’s Rule of Signs
Copyright © 2011 Pearson Education, Inc. Slide 3-6
3.3
Theorem on BoundsSuppose that P(x) = 0 is a polynomial with real coefficients
and a positive leading coefficient, and synthetic division
with x – c is performed. If c > 0 and all terms in the bottom row are nonnegative,
then c is an upper bound for the roots of P(x) = 0. If c < 0 and the terms in the bottom row alternate in sign,
then c is a lower bound for the roots of P(x) = 0.
Bounds on the Roots