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I highly recommend chapters 1, 2, 5, 7, 9, 11 of the book Calcul mathématique avec Sage, also available in an English version.

Note that to find roots of a quotient of polynomials, you can use the numerator only, and forget about the denominator.

There are well-known methods to count, find, isolate roots of polynomials. Some of them are implemented in Sage.

See these pages in SageMath's reference manual:

and you can also search the web for "root finding algorithms" and "root isolation".