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

Upper Bound

If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f(x) = 0.

Note that two things must occur for c to be an upper bound.  One is c > 0 or positive.  The other is that all the coefficients of the quotient as well as the remainder are positive.

Lower Bound

If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0.  Special note that zeros can be either positive or negative.  

Note that two things must occur for c to be a lower bound.  One is c < 0 or negative.  The other is that successive coefficients of the quotient and the remainder have alternating signs.

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