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This calculator computes complex and real roots for any cubic polynomial. It applies the Lin-Bairstow algorithm which iteratively solves for the roots starting from random guesses for a solution. The calculator is designed to solve for the roots of a cubic polynomial with the form:

    x3 + ax2 + bx + c = 0

The program is operated by entering the coefficients for the cubic polynomial to be solved, selecting the rounding option desired, and then pressing the Calculate button. All entries are cleared by pressing the Clear button. If the value of c is zero (which means that one root is zero), the program returns an error message:

cannot solve

In this case, the cubic polynomial can be reduced to a quadratic which is easily solved. It is possible for the initial random guesses used by the algorithm to cause it to be unstable; the above error message will result in this instance. Each time the algorithm is started, a new set of initial random guesses will be generated - another trial may result in a solution

0 = x3 + x2 + x +

Apply rounding
No rounding

x1 = + j

x2 = - j

x3 =


A polynomial, P(x), has a factor of (x - r) if and only if P(r) = 0. Then r is said to be a zero of the polynomial.

Every cubic polynomial, P(x), has a factorization of the form:

    P(x) = (x - r1)(x - r2)(x - r3) = 0
where the roots, ri, can be duplicates.

If P(x) has real coefficients (as in this calculator), and if x is a complex zero of P(x), then the complex conjugate of x is also a zero of P(x). A cubic polynomial can have three real zeros, or one real zero and one pair of complex zeros

Copyright 2004, Stephen R. Schmitt