Civil Engineering Reference
In-Depth Information
Find possible positive roots for
f
(
x
)
=
0:
3
2
6160
x
−+−=
x
x
1
2
3
=
3 sign changes
Since there are three sign changes, there is a maximum of three positive
roots. Three positive real roots exist or one positive real root plus two
imaginary roots.
Find possible negative roots by rewriting the function for
f
(−
x
) = 0:
−
()
−
()
+
(
−=−− −−=
−− −−=
3
2
3
2
x
6
x
11
x
6 6160
x
x
x
3
2
x
6160
x
x
0
0
0
=
0 sign changes
Notice the signs of all the odd powers reverse while the signs of the even
powers remain unchanged. Count the number of sign changes,
n
. This
number is the maximum possible negative roots. Since there is no sign
change, zero negative roots exist.
Possible complex roots:
Complex roots appear in conjugate pairs. Therefore, either zero or two
complex roots exist. In this example the roots are
x
= 1, 2, 3.
Example 1.2 Descartes' rule
Find the possible number of positive, negative, and complex roots for the
following polynomial:
3
2
760
x
−+=
x
Find possible positive roots for
f
(
x
) = 0:
3
2
760
x
−+=
x
1
2
=
2 sign changes
Since there are two sign changes, there is a maximum of two positive
roots. Two or zero positive real roots exist.
