POLYNOMIAL EQUATIONS

The schemes developed until now can be used also to find a root of polynomial equations. The main disadvantage of these schemes is, they can give only one root of  f(x) = 0   in  one iterative process. To find any other root one  supposed  to  change  the  initial  value  and  repeat  the  iterative  process. However, this does not guarantee a second root and in some situations the iterative process can even diverge.

So in this  section we will concentrate on the schemes which can give all  the  roots  of  the given  polynomial  Pn(x) = 0  where n   is the degree of the polynomial.

In a polynomial if a +ve sign follows a +ve sign or a -ve sign follows a -ve sign then a continuation of sign is said to occur. However, if a +ve sign follows a -ve sign or vice-verca then a change of sign is said to occur.

    Descarte's Rule of signs :

The number of +ve real roots of Pn(x) = 0 cannot exceed the number of sign changes in P n(x) and the number of -ve real roots of Pn (x) = 0 cann't exceed the number of sign changes in Pn(-x).

For example P5(x) = x 5 + 2x4 - x3+x2 - x + 2 = 0 has four sign changes and hence it can have at most four +ve real roots and one -ve real root

This rule can give only an Upper hand for the number of +ve and -ve real roots but cannot give the exact number. This can be obtained from Sturm's theorem.

If f(x) be a given Polynomial of degree 'n' and let f1(x) is its first derivative. Denote by f2 the remainder of f(x) divided by f1(x) with the opposite sign and f3 is the remainder of f1(x) divided by f2(x) with opposite sign and so on, until a constant is arrived.

    i.e., f(x)  = Pn (x)
           f1(x) = f '(x)
           f2(x) = -[rem(f(x)/f01(x))]
           f3(x) = -rem(f1(x)/f2(x))

    and so on

Then the sequence of f, f1, f2 . . ., fn is called Sturm function or Sturm sequence.

       
    Sturm theorem :

The number of real roots of the equation f(x) = 0 at [a,b] equal the difference between the number of changes of sign in the Sturm sequence at x = a and x= b, provided that f(a) ¹ 0 and f(b) ¹ 0.

Since any nth degree polynomial has exactly n roots, the number of complex roots equals (n - number of real roots), when a real root of multiplicity r is to be counted r times. If the coefficient of the polynomial are real then the complex roots are  a ± ib and hence total no of complex roots are even.