Question

Locating Zeros of Polynomial Function:

Describe how you find an upper bound and a lower bound for the zeros of a polynomial function

Answer 1

Use synthetic division to find the values of the polynomial function for consecutive integers. An integer that produces no sign change in the quotient and the remainder is an upper bound. To find a lower bound of a function, find an upper bound for the function of -x. The lower bound is the negative of the upper bound for the function of -x.

That is the right answer as I just finished the 10 practice problems on E2020

Answer 2

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.

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The division can be either synthetic division or long division.

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

• Divide all coefficients by that of the highest-degree term
• Find the absolute value of the result
• Choose the maximum of those results and increase it by 1

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.