Locating zeros of polynomial function: describe how you find an upper bound and a lower bound for the zeros of a polynomial function

The value p is an upper bound on the real roots of the polynomial if division of the polynomial by (x-p) results in quotient terms that all have positive coefficients.

The value q is a lower bound on the real roots of the polynomial if division of the polynomial by (x-q) results in quotient terms that alternate signs.

The division can be either synthetic division or long division.

An upper bound on the magnitude of the roots can be found this way:

There are various refinements, including finding the sum of the ratios of successive coefficients, or taking increasing roots of the ratios found above, then doubling the maximum of those.

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