3.4 Zeros of Polynomial Functions
5
3.4 Zeros of Polynomial Functions 1 1 1 0 0 If () hasintegercoefficientsand (w here isreduced)isarationalzero, then isa factoro The RationalZero Theorem fthe constantterm and isa factorofthe lead n n n n p p q q fx ax a x ax a p a q ing coefficient . n a
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3.4 Zeros of Polynomial Functions. Properties of Polynomial Equations. Properties of Polynomial Equations contd. Descartes’s Rule of Signs. Descartes’s Rule of Signs contd. - PowerPoint PPT Presentation
Transcript of 3.4 Zeros of Polynomial Functions
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.
