Section 4.5 Class Notes
The Real Zeros of a Polynomial Function
Example 1: Find the real zeros of the polynomial function
5
4
( )
4
12
3
f x
x
x
x
=
+
 
How many zeros
could
this polynomial have?
Number of Real Zeros Theorem
A polynomial function cannot have more real zeros than its
degree.
How many positive zeros could this polynomial have?
Descartes’ Rule of Signs
If
f
is a polynomial written in standard form, then the number
of positive zeros either equals the number of variations in
sign of the nonzero coefficients of
f (x)
or else equals that
number less an even integer.
How many negative zeros could this polynomial have?
Descartes’ Rule of Signs
If
f
is a polynomial written in standard form, then the number
of negative zeros either equals the number of variations in
sign of the nonzero coefficients of
f(x)
or else equals that
number less an even integer.
What are all the possible rational zeros of this polynomial?
Rational Zeros Theorem
If f is a polynomial with degree 1 or higher, with integer
coefficients and has the form
1
1
1
0
0
( )
,
0,
0
n
n
n
n
n
f x
a x
a
x
a x
a
a
a


=
+
+
+
+
L
then the set of fractions formed by dividing all factors of
0
a
by the all factors of
n
a
will contain all possible rational
zeros of
f
.
(Note that not all such fractions will be zeros of
f
.
It may be that none of these fractions is a zero of
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 '10
 Bergthold,Trisha
 Algebra, Factor Theorem, Rational Zeros Theorem, Multiplication

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