HP 33s Polynomial Root Finder, 15–20, on, g@, g@, g@, g@ ) , where b2 = – a2, Displays next value, Displays next value, Displays next value

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Polynomial Root Finder

This program finds the roots of a polynomial of order 2 through 5 with real coefficients. It calculates both real and complex roots.

For this program, a general polynomial has the formxn + an–1xn–1+ ... + a1x + a0 = 0

where n = 2, 3, 4, or 5. The coefficient of the highest–order term (an) is assumed to be 1. If the leading coefficient is not 1, you should make it 1 by dividing all the coefficients in the equation by the leading coefficient. (See example 2.)

The routines for third– and fifth–order polynomials use SOLVE to find one real root of the equation, since every odd–order polynomial must have at least one real root. After one root is found, synthetic division is performed to reduce the original polynomial to a second– or fourth–order polynomial.

To solve a fourth–order polynomial, it is first necessary to solve the resolvant cubic polynomial:

y3 + b2y2 + b1y + b0 = 0where b2 = – a2

b1 = a3a1– 4a0

15–20Mathematics Programs