Polyroots is abundantly reported to fail. For sure, it does not understand
the canonical cubic, from which canonical the 3 roots are extracted and
reconstructed via Lagrange roots form. Smath is slightly pregnant to plot
Lagrange complex roots [only some months of pregnancy].

In Process Control & Instrumentation, the PI & PID modes have only a vector
of 3 coefficients, that represent the canonical cubic involved in the Laplace
solution. Polyroots fail to solve the canonical.
The cubic canonical is reconstructed via the Lagrange roots factor.
These two companions are inseparable birds.

Note that roots of orthogonal polynomial aren't done via polyroots, they
have their own solution [see Abramowitz & Stegun]. Mathcad 11 OK.

If you read the attached document, you will discover that if you want to
use polyroots on the example, you will have to teach user to split 3 cats
to get 4 cats. No problem to get 4 delicatessen out of 3 cats, but you have
no more cat.

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Cubic solver [Types 1, 2].sm (58 KiB) downloaded 166 time(s).