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This calculator computes complex and real roots for any quintic 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 quintic polynomial with the form:

    x5 + ax4+ bx3 + cx2 + dx + e = 0

The program is operated by entering the coefficients for the quintic 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 d is zero (which means that one root is zero), the program returns an error message:

cannot solve

In this case, the quintic polynomial can be reduced to a quartic which cannot be solved using this calculator; try the Quartic Root Calculator. 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 = x5 + x4 + x3 + x2 + x +

Apply rounding
No rounding

x1 = + j

x2 = - j

x3 = + j

x4 = - j

x5 =


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

    P(x) = (x - r1)(x - r2)(x - r3)(x - r4))(x - r5) = 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 quintic polynomial can have five real zeros, or three real zeros and one pair of complex zeros, or one real zero and two pairs of complex zeros

Copyright 2004, Stephen R. Schmitt