# Rule Of Signs

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Thm. The number of positive roots is at most the number of sign
changes in the polynomial's coefficients.
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DISCLAIMER: Discard zero coefficients.

For negative roots, use p(-x). (maximum)
For nonreal roots, use n-(positives)-(negatives). (minimum)

Because of the conjugacy, the number of nonreal roots is always
even. So if the number of sign changes is zero or one, the number
of positive roots equals the number of sign changes.

The reason is this. Let the maximum number of positive roots be n.
Then the possibilities are n, n-2, n-4, ... because of the
possibilities of nonreal roots. So if n is either 0 or 1, that is
the exact number of positive roots.


## Example

For p(x) = x^3 + x^2 - x - 1, ++-- so it has exactly one positive
root. (+ "+-" -)

p(-x) = -x^3 + x^2 + x - 1, -++- so it has two or zero negative
roots. ("-+" "+-")

So the possibilities are [3 real (+--) roots] or [1 real (+) and 2
nonreal roots].

For q(x) = x^3 - 1, +- so it has exactly one positive root.

q(-x) = -x^3 - 1, -- so it has exactly zero negative root.

So the exact number of nonreal roots of q(x) is 3-1-0=2. You would
already know that the roots are 1, w, w^2. These form the finite
field F_3.