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