Root Multiplicity and Polynomial Functions
Date: 11/16/97 at 01:09:55 From: alex rubinshteyn Subject: Root Multiplicity and Polynomial Functions What effect does multiplicity have on a polynomial function? Here's an example of what I mean by multiplicity (x+1)(x-2)^2 where -1 has a multiplicity of 1 and 2 of 2. I can't figure out what effects multiplicities have, although odd multiplicities appear to make the function parallel to the x axis for a while, like x(x-2)^9. But 3 is not parallel to the x axis, nor is 1. The higher the multiplicity, the farther the function appears to travel along the x axis, but only up to a point. Can someone please help?
Date: 11/16/97 at 09:59:25 From: Doctor Jerry Subject: Re: Root Multiplicity and Polynomial Functions Hi Alex, You have done well in thinking about the effect on the graph of multiple roots. If y=q(x)*(x-a)^k, where q(x) is the rest of the polynomial we are considering, if k is large, then the factor (x-a)^k will be very small within 1 of a. This means that the graph will be quite flat in the interval (a-1,a+1). Of course, if k is only 1 or 2, the flatness is limited to a short interval about a. When k is even, the factor (x-a)^k doesn't change sign as x moves from the left of a to the right of a (assuming that q(x) doesn't have roots very close to a). So, the graph will be tangent to the x-axis. For odd k, the graph crosses the x-axis, even though it may be quite flat. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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