Civil Engineering Reference
In-Depth Information
• The coefficients (
a
0
,
a
1
,
a
2
…,
a
n
-1
,
a
n
) are real numbers.
• There will be at least one real root if
n
is an odd integer.
• It is possible that equal roots exist.
• When complex roots exist, they occur in
conjugate pairs
. For
example:
xuvi uv
=±=± −1
1.3
DEScARtES' RuLE
Descartes' rule is a method of determining the maximum number of pos-
itive and negative real roots of a polynomial. This method was published
by René Descartes in 1637 in his work La Géométrie (Descartes 1637).
This rule states that the number of positive real roots is equal to the num-
ber of sign changes of the coefficients or is less than this number by an
even integer. For positive roots, start with the sign of the coefficient of
the lowest (or highest) power and count the number of sign changes from
the lowest to the highest power (ignore powers that do not appear). The
number of sign changes proves to be the number of positive roots. Using
x
= 1 in evaluating
f
(
x
) = 0 is the easiest way to look at the coefficients.
For negative roots, begin by transforming the polynomial to
f
(
−x
) = 0.
The signs of all the odd powers are reversed while the even powers remain
unchanged. Once again, the sign changes can be counted from either the
highest to lowest power, or vice versa. The number of negative real roots
is equal to the number of sign changes of the coefficients, or less than by
an even integer. Using
x
=
-1
in evaluating
f
(
x
) = 0 is the easiest way to
look at the coefficients.
When considering either positive or negative roots, the statement
“less than by an even integer” is included. This statement accounts for
complex conjugate pairs
that could exist. Complex conjugates change the
sign of the imaginary part of the complex number. Descartes' rule is valid
as long as there are no zero coefficients. If zero coefficients exist, they are
ignored in the count. Also, one could find a root and divide it out to form a
new polynomial of degree “
n
- 1” and apply Descartes' rule again.
Example 1.1 Descartes' rule
Find the possible number of positive, negative, and complex roots for the
following polynomial:
3
2
6160
x
−+−=
x
x
