ROOTS OF EQUATIONS

The purpose of calculating the roots of an equation is to determine the values of x for which holds:

f (x) = 0 (28)

The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum, eigenvalues of matrices, solving systems of linear differential equations, etc ...

The determination of the solutions of equation (28) can be a very difficult problem. If f (x) is a polynomial function of grade 1 or 2, know simple expressions that allow us to determine its roots. For polynomials of degree 3 or 4 is necessary to use complex and laborious methods. However, if f (x) is of degree greater than four is either not polynomial, there is no formula known to help identify the zeros of the equation (except in very special cases).

There are a number of rules that can help determine the roots of an equation:

• Bolzano's theorem, which states that if a continuous function, f (x) takes on the ends of the interval [a, b] values of opposite sign, then the function accepts at least one root in that interval.

• In the case where f (x) is an algebraic function (polynomial) of degree n and real coefficients, we can say that will have n real roots or complex.

• The most important property to verify the rational roots of an algebraic equation states that if p / q is a rational root of the equation with integer coefficients:

then the denominator q divides the coefficient a and the numerator p divides the constant term a0.

Example: We intend to calculate the rational roots of the equation:

3x3 3x2 - x - 1 = 0

First, you make a change of variable x = y / 3:

and then multiply by 32:

3y2 y3-3y = -90

with candidates as a result of the polynomial are:

Substituting into the equation, we obtain that the only real root is y = -3, that is to say, (which is also the only rational root of the equation). Logically, this method is not as effective, so we can serve only as guidelines.

Most of the methods used to calculate the roots of an equation are iterative and are based on models of successive approximations. These methods work as follows: from a first approximation to the value of the root, we determine a better approximation by applying a particular rule of calculation and so on until it is determined the value of the root with the desired degree of approximation.