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Graphing Polynomial Functions in Factored Form

Date: 07/07/2004 at 13:42:59
From: Shannon
Subject: Multiplicities of Polynomial Functions

Why does even multiplicity cause a graph to touch an intercept and odd 
multiplicity cause a graph to cross it?



Date: 07/07/2004 at 14:03:34
From: Doctor Vogler
Subject: Re: Multiplicities of Polynomial Functions

Hi Shannon,

Thanks for writing to Dr Math.  Suppose we have a polynomial

  f(x) = (x - r)^m * (other terms)

with a root r of multiplicity m.  So the other terms are not near zero
when x is near r.  That is, if you look close enough to r, the other
terms will be close to some nonzero number, either positive or 
negative.  Or, said differently, the limit as x approaches r of those
other terms is some nonzero real number.  If it was zero, then we
could factor out another x - r from those other terms, and the root r
would have multiplicity m + 1.

So then when x = r, f(x) = f(r) = 0.  That causes the graph to touch
the x-axis at (r,0).  Now when x is just larger than r (that is, just
to the right of r), then x-r will be positive, and (x-r)^m will also
be positive.  So f(x) will have the same sign as the product of those
other terms.  But when x is just smaller than r, then x-r will be
negative, and (x-r)^m might be negative and might be positive.  In
fact, (x-r)^m will be negative if m is odd, and (x-r)^m will be
positive if m is even.  So if m is even, then f(x) will have the same
sign as those other terms, which means it approaches the 0 at f(r) but
has the same sign on both sides of r, so it touches and does not 
cross.  But if m is odd, then f(x) will have the opposite sign as
those other terms, which means it has a different sign on each side of
r, so it crosses the axis.

Does that make sense?  If you have any questions about this or need
more help, please write back, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Functions
High School Polynomials

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