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how to find the zeroes and their multiplicities of a polynomial without factoring?

for example: x^5 - x^4 - 2x^3 + 2x^2 + x - 1
zeroes and multiplicities without needing to factor the whole equation
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I'd start by gaining as much knowledge as I can and sketching what I find on a graph.

1. The polynomial is of degree 5 so will cross the x axis a maximum of 5 times.
2. The order is odd (edit: and the largest term has a positive coefficient) so I am expecting a function with large +x, large +y, large -x, large -y.
3. f(0)=-1 so I can expect at least one crossing of the x axis in x>0.
4. Try some obvious (and easy) points x=1, x=-1.
5. Evaluate the function at a few points to be able to zero in on places where it might cross the x axis and to get enough data to plot the graph.
6. Try integer and likely values near where I feel the graph may cross based on my rough plot.
7. Use Newton's method to find non-integer points, with starting values from my rough plot.
8. At every stage in this process, resist the urge to factor the function.
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There are a number of ways to do this with calculus, but I imagine that it may be a bit "high brow" of a way of doing things. You can compute where the local maxima/minima are, then use that information to pinpoint where roots could be. At that point you can guess and check.

For your example, though, there is a very easy factorization (which takes far less effort than other approaches). terms 1+2, 3+4, and 5+6 together have a  factor of x-1, so you get (x\^4 - 2x\^2 + 1)(x-1). The first term in this factorization is just (x\^2 - 1)\^2, which further factors into (x-1)\^2 \* (x+1)\^2. Thus you get a complete factorization of (x-1)\^3 \* (x+1)\^2, so this polynomial has a root of multiplicity 3 at 1, and a root of multiplicity 2 at -1.
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