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/