Civil Engineering Reference
In-Depth Information
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, 1 +
i
,
1
-
i.
It
should be noted that the existence of complex conjugate pairs cannot be
readily known. Examples 1.1 and 1.3 had the same number of sign change
count, but the latter had a complex pair of roots.
Example 1.4
Descartes' rule
Find the possible number of positive, negative, and complex roots for the
following polynomial:
xx x
3
−+=
2
20
Find possible positive roots for
f
(
x
)
=
0:
xx x
3
−+=
2
20
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.
Find possible negative roots by rewriting the function for
f
(
-x
) = 0:
−
()
−
()
+
()
=− −−=
−−−=
3
2
x
x
2
x
x
3
x
2
2
x
0
xx x
3
2
20
0
0
=
0 sign changes
Count the number of sign changes,
n
. 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
=
0, 1 +
i
, 1 -
i.
The existence of zero as a root could have been discovered by noticing
that there was not a constant term in the equation. Therefore, dividing the
equation by
x
yields the same as
x
=
0
.
