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