Chapter 2

2.2 Polynomial Equations

Let f(x) be a polynomial of degree and with real or complex coefficients. On equating it to zero we get the equation: f(x)=0.
The problem of solving an equation consists in finding all its roots, i.e. the roots of the polynomial f(x).
If the degree of f(x) is n then the corresponding equation is said to be of degree n.

Here some theorems, relevant  to the roots of polynomials, which are useful for the solving of polynomial equations are presented.  The proofs of these theorems are deliberately omitted since they can be found in any book relevant to polynomial equations.

Theorem 1: The Fundamental Theorem of Algebra

Every polynomial f(x) of degree , with complex coefficients has at least one complex solution.

Theorem 1 guarantees the existence of a solution for every polynomial equation of degree , but it does not inform as about the number of solutions.

Theorem 2

The number p is a root of the polynomial f(x) if and only if, there is a polynomial such that
.

At this point it must be noted that:
If p is also a root of  then there is polynomial such as
therefore ,
and as it is clear it is possible to end up with an equation of the form.
Then we say that p is a root of multiplicity k and is counted as k roots equal to p.

The next theorem refers to the number of solutions of a polynomial equation.

Theorem 3

Every polynomial f(x) with complex solutions, of degree has exactly n solutions, were each solution is counted as many times as its multiplicity.

From theorem 3, we conclude that if  are the n solutions of
then f(x) gets the form :

Theorem 4: The Vieta Formulas

If are the roots of a polynomial f(x) of degree , then

Theorem 5

If a polynomial, which is not identically vanishing, with real coefficients has as a root the complex number z = a + bi ( ) then the conjugate of z, , is also a root of the polynomial.

From Theorem 5 we can conclude the following:

• The number of complex roots of a polynomial with real coefficients, is an even number
• A polynomial with real coefficients, whose degree is an odd number has at least one real solution.
Theorem 6

If a polynomial f(x) with rational coefficients, which is not identically vanishing, has as a root the irrational number irrational, then it will also have as a root the number
.
Similar conclusions to those of Theorem 5 can be derived.

Theorem 7

If  a polynomial with integer coefficients has as a root the rational number , then k is a divisor of and  a divisor of .

Theorem 8

If the polynomial  with integer coefficients has as a root the integer number k, then k  is a divisor of .