3.4 Zeros of Polynomial Functions

11 1 0

0

If ( )

has integer coefficients and (where

is reduced) is a rational zero, then is a

factor o

The Rational Zero Theorem

f the constant term and is a

factor of the lead

n nn n

p pq q

f x a x a x a x a

p

a q

ing coefficient .na

Properties of Polynomial Equations

1. If a polynomial equation is of degree

, then counting multiple roots separately,

the equation has roots.

n

n

Properties of Polynomial Equations contd.

2. If is a root of a polynomial equation

( 0), then the nonreal complex number

is also a root. Nonreal complex roots,

if they exist, occur in conjugate pairs.

a bi

b

a bi

Descartes’s Rule of Signs1 1

1 1 0Let ( )

be a polynomial with real coefficients.

n nn nf x a x a x a x a

1. The number of of

is either equal to the number of sign changes

of ( ) or is less than the number by an even

integer. If there is only one variation in sign,

posi

ther

tive real z

e is exactl

eros

y one

f

f x

positive real zero.

Descartes’s Rule of Signs contd

1 11 1 0Let ( )

be a polynomial with real coefficients.

n nn nf x a x a x a x a

2. The number of of

is either equal to the number of sign changes

of ( ) or is less than the number by an even

integer. If ( ) has only one variation in

negative

sign,

the

real

n has exactly

zeros f

f x

f x

f

one negative real zero.