Graphing Polynomial Functions in Factored FormDate: 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/ |
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