Civil Engineering Reference
In-Depth Information
Find possible negative roots by rewriting the function for
f
(
-x
) = 0:
−
()
−
()
+=−− +=
−− +=
3
2
3
2
x
7 6 760
760
x
x
x
3
2
x
x
0
1
=
1 sign change
Again, 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 one sign
change, one negative root exists.
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 1.3
Descartes' rule
Find the possible number of positive, negative, and complex roots for the
following polynomial:
3
2
3460
x
−+−=
x
x
Find possible positive roots for
f
(
x
) = 0:
3
2
3460
x
−+−=
x
x
1
2
3
=
3 sign changes
Since there are three sign changes, there is a maximum of three positive
roots. Three or one positive real roots exist.
Find possible negative roots by rewriting the function for
f
(
-x
) = 0:
−
()
−
()
+
(
−=−− −−=
−− −−=
3
2
3
2
x
3
x
4
x
6
x
3 460
x
x
3
2
x
3460
x
x
0
0
0 =
0 sign changes
Count the number of sign changes. Since there is no sign change, zero
negative roots exist.
