Mathematical Fundamentals
119
solutions
zeros
roots
Fundamental Theorem of
Algebra
The second-order polynomials, when represented in the x - y plane, produce
a class of curves called parabolas, while third-order polynomials generate cubic
curves.
We call solutions, or zeros, or roots of a polynomial those values of the
independent variable that produce a zero value of the dependent variable. For
second and third-order polynomials there are formulas to derive the zeros in
closed form. Particularly important is the formula for second-order polynomials:
ax
2
+ bx + c = 0
(7)
x =
-b
b
2
- 4ac
2a
.
(8)
As it can be easily seen by application of (8) to the polynomial x
2
+ 1,
the roots of a real-coefficient polynomial are real numbers. This observation
was indeed the initial motivation for introducing the complex numbers as an
extension of the field of real numbers.
The Fundamental Theorem of Algebra states that every n-th order real-
coefficient polynomial has exactly n zeros in the field of complex numbers, even
though these zeros are not necessarily all distinct from each other. Moreover,
the roots that do not belong to the real axis of the complex plane, are couples
of conjugate complex numbers.
For polynomial of order higher than three, it is convenient to use numerical
methods in order to find their roots. These methods are usually based on some
iterative search of the solution by increasingly precise approximations, and are
often found in numerical software packages such as Octave.
In Octave/Matlab a polynomial is represented by the list of its coefficients
from a
n
to a
0
. For instance, 1 + 2x
2
+ 5x
5
is represented by
p = [5 0 0 2 0 1]
and its roots are computed by the function
rt = roots(p) .
In this example the roots found by the program are
rt =
-0.87199 + 0.00000i
0.54302 + 0.57635i
0.54302 - 0.57635i
-0.10702 + 0.59525i
-0.10702 - 0.59525i
and only the first one is real. If the previous result is saved in a variable rt,
the complex numbers stored in it can be visualized in the complex plane by the
directive
axis([-1,1,-1,1]);
plot(real(rt),imag(rt),'o');
and the result is reported in fig. 5.
It can be shown that the real-coefficient polynomials form a commutative
ring with unity if the operations of sum and product are properly defined. The
sum of two polynomials is a polynomial whose order is the highest of the orders
of the operands, and having coefficients which are the sums of the respective
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