APPROXIMATE SOLUTION OF A PARAMETRIC EQUATION OF THE FIFTH DEGREE

Abstract

The problem of finding the roots of high-degree polynomials is complex and generally unsolvable. However, in a number of special cases, the roots can be found. The article proposes an original approach to finding the roots of a partial fifth-degree polynomial containing a parameter as a free term. An attempt to find the roots of a parametric fifth-degree polynomial by representing this polynomial as a product of third- and second-degree polynomials, with the subsequent compilation of a system of equations for finding the coefficients of third- and second-degree polynomials, leads to very cumbersome equations, the complexity of solving which is very high. Therefore, an approach is proposed, the idea of which is that the roots are first found for fixed values of the parameter. Then, by setting a small increment to the parameter value, an analysis is carried out for changes in the values of the roots of the polynomial. This becomes possible due to the fact that the parameter is a free term of the polynomial and its increment leads to a shift in the polynomial graph along the vertical axis. This approach allows finding approximate values of roots without using iterative numerical methods.