Let be a polynomial of order n-1 with m real, distinct zeros.
a polynomial of order k-1 with j real, distinct zeros.
What can we say about the division

Notes: (TO DO: Break this up into several worksheets).
What happens if I write ?
At a real zero of escapes to . That is, has a vertical asymptote at each real, distinct zero of q(x). Now, I will impose a
Condition: Assume that that don't share a common zero in the interval a<x<b.
Then the real zeros of r(x) in a<x<b are exactly the real zeros of p(x)
The vertical asymptotes of r(x) in a<x<b are exactly the real zeros of q(x)
I will use this class of functions to hunt for the roots of a polynomial. In words, we can partition the x-range into distinct intervals using q(x), while preserving the zeros of p(x). The behavior of r(x) across the asymptotes gives us information about the number of zeros in each interval.
Example: Let q(x) = p'(x). Between two successive zeros of p(x), the curve must turn around and come back (Rolle's theorem): p'(x) must have at least one zero between any two real zeros of p(x). That is, at least one asymptote separates each real distinct zero of r(x)=p(x)/p'(x). The roots can be isolated.
How can we use this information? Consider a sequence of polynomials of decreasing order, beginning with
.
If I write , the zeros are isolated, but if the order of f2 >3, I still cannot easily find them. So I will keep dividing. How do I go about this divison?
How shall I define ?
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