Let f(x) be a polynomial of degree and
with real or complex coefficients. On equating it to zero we get the equation:
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.
The number p is
a root of the polynomial f(x) if and only if, there is a polynomial such
At this point it must be noted that:
If p is also a root of then there is polynomial such as
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.
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
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:
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.
If , a polynomial with integer coefficients has as a root the rational number , then k is a divisor of and a divisor of .
If the polynomial , with integer coefficients has as a root the integer number k, then k is a divisor of .