Roots of Polynomials with UltimaCalc

Roots of Polynomials with UltimaCalc

UltimaCalc contains a polynomial root calculator that can calculate the roots of a polynomial up to the 10th order. The roots are the values of x for which a polynomial in x has the value 0. Thus, these values of x are also called zeroes of the polynomial.

Traditionally, finding these roots or zeroes has been a challenge. Explicit formulas exist only for the direct calculation of the roots of polynomials up to the 4th order, though for 3rd and 4th order these formulas are complicated. It can be proved that no formula exists for the solution of fifth or higher order polynomials. Finding polynomial roots is a useful method to help when factorising a polynomial.

Polynomials - Higher Orders

For higher order polynomials, various strategies have been devised. If the coefficients are integers and the order is low, intelligent guesswork can sometimes suffice. Another approach is to search for regions within which the value of the polynomial changes sign, and then zooming in on the zero. Such a strategy cannot guarantee to be always successful in finding all the roots of a polynomial, especially when they are close together.

UltimaCalc uses the Jenkins and Traub method which, although far too complicated to describe here, is of proven reliability. The precise locations of the roots (or even their existence) can be very sensitive to the coefficients of the polynomials and to accumulated rounding errors. The 38 digit precision to which UltimaCalc works helps to minimise these latter errors.

The image below shows an easy task - finding the roots of the cubic polynomial:

x3 + 2.x2 - x - 2

Roots of a cubic polynomial

The results show that this polynomial can be factorised as:

(x + 2)(x + 1)(x - 1)

The button marked 'Copy' copies the calculation of the polynomial roots to the Windows clipboard. In the above example, this results in the following text:

Polynomial Roots

Roots of Polynomial of order 3:

A3 = 1

A2 = 2

A1 = -1

A0 = -2

The roots are:




The 'Log' button writes the calculation to a log file. The 'Open', 'Save', and 'Save As' buttons allow calculated polynomial roots to be saved, along with notes, so that the calculation can be repeated later, perhaps after changing some of the coefficients.

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