HP 50g Graphing Calculator - Manual (Page 213)

Page 6-7
All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (-0.766,
0.632), (-0.766, -0.632).
Generating polynomial coefficients given the polynomial's roots
Suppose you want to generate the polynomial whose roots are the numbers [1,
5, -2, 4]. To use the calculator for this purpose, follow these steps:
‚Ϙ˜
@@OK@@ Select solve poly...
˜„Ф1‚н5
‚н2\‚н 4
@@OK@@ Enter vector of roots
@SOLVE@ Solve for coefficients
Press ` to return to stack, the coefficients will be shown in the stack.
Note: Recall that complex numbers in the calculator are represented as
ordered pairs, with the first number in the pair being the real part, and the
second number, the imaginary part. For example, the number (0.432,-0.389),
a complex number, will be written normally as 0.432 - 0.389i, where i is the
imaginary unit, i.e., i
2 = -1.
Note: The fundamental theorem of algebra indicates that there are n solutions
for any polynomial equation of order n. There is another theorem of algebra
that indicates that if one of the solutions to a polynomial equation with real
coefficients is a complex number, then the conjugate of that number is also a
solution. In other words, complex solutions to a polynomial equation with real
coefficients come in pairs. That means that polynomial equations with real
coefficients of odd order will have at least one real solution.