WHY do x-intercepts with different multiplicities behave differently?

howiejackson14

New member
Joined
Nov 29, 2014
Messages
3
WHY do x-intercepts with different multiplicities behave differently? I know that odd multipicites go through the x-axis and that even multiplicities bounce off the x-axis, but WHY? I NEED HELP Understanding this!!! Could this be explained algebraically and not with calculus reasoning??
 
Last edited by a moderator:
I know that odd multipicites go through the x-axis and that even multiplicities bounce off the x-axis, but WHY? I NEED HELP Understanding this!!! Could this be explained algebraically and not with calculus reasoning??
Suppose we have a zero at x=A with multiplicity of m. Then the function f(x) can be written as (x-A)m g(x). If g(x) is positive (negative) for x just smaller than A, then it is still positive (negative) for x just greater than A (assuming, of course, continuity, etc.).

However, suppose m is odd. Then (x-A)m changes sign as x passes through A and since g(x) remains the same sign, f(x) changes sign, i.e. goes through the x-axis.

If m is even then ...
 
Top