Friday, September 25, 2009

polynomials - bounded roots


This sketch examines the test for an upper or lower bound of real zeros for a polynomial function. The test uses Descartes' Signs Rule that gives a maximum number of zeros according to the number of times the polynomial terms change signs. The user can alter the quartic polynomial, which is defined by five independent points. (There is also a tool in the sketch for constructing a quartic polynomial from five points).

This sketch shows the synthetic division of f(x) by (x-k). Particularly interesting is the chance to observe that some bounds will not be recognized as such by this test. Under what conditions will the test fail? How can we employ this idea for a polynomial with a negative leading coefficient? What must be true in order for the the least upper bound to equal the greatest root? How do complex roots affect the result? Why is there a restriction that k ? 0 in order to demonstrate that k is an upper bound? Could an adaptation be made to allow us to test a negative k as an upper bound? Does a horizontal translation of the function affect the difference between the least upper bound and the greatest root?

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