How Does the Derivative Change the Multiplicity of a Zero?

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In summary, if a function f(t) has a zero of multiplicity n, it can be written as (t-t_0)^n * g(t), where g(t_0) is not equal to 0. When taking the derivative of f(t), the multiplicity of the zero at t_0 will be exactly n-1. This can be proven by writing f'(t) = (t-t_0)^(n-1)k(t) where k(t) = ng(t) + (t-t_0)g'(t) and k(t_0) = ng(t_0) which is not equal to 0.
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ehrenfest
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Homework Statement


How do you show that if f(t) has a zero t_0 of multiplicity n, then f'(t) has a zero at t_0 of multiplicity EXACTLY n -1?

I can prove that the multiplicity is at least n-1 but I am having trouble with the exactly part...


Homework Equations





The Attempt at a Solution

 
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  • #2
if f(t) has a root of multiplicity n, then we can write f(t) = (t - t_0)^n g(t) where g(t_0) != 0. Then f'(t) = (t-t_0)^(n-1)k(t) where k(t) = ng(t) + (t-t_0)g'(t) and k(t_0) = ng(t_0) != 0, so f' has a root of multiplicity n-1.
 

FAQ: How Does the Derivative Change the Multiplicity of a Zero?

What are multiplicities of zeroes?

Multiplicities of zeroes refer to the number of times a polynomial function intersects the x-axis at a specific value. These multiplicities can indicate the behavior of the graph near that point.

How do I find the multiplicity of a zero?

To find the multiplicity of a zero, you need to factor the polynomial and look at the exponent of the factor corresponding to the zero. The exponent represents the multiplicity of that zero.

What do multiplicities of zeroes tell us about the graph of a polynomial function?

The multiplicities of zeroes can tell us about the behavior of the graph near those points. For example, a zero with an odd multiplicity will cross the x-axis, while a zero with an even multiplicity will bounce off the x-axis.

Can a polynomial function have multiple zeroes with the same value but different multiplicities?

Yes, a polynomial function can have multiple zeroes with the same value but different multiplicities. This means that the graph will intersect the x-axis at that value multiple times, with each intersection having a different behavior depending on the multiplicity of the zero.

How can I use multiplicities of zeroes to help me graph a polynomial function?

Multiplicities of zeroes can help you determine the shape and behavior of the graph near each zero. You can use this information to plot points on the graph and make accurate sketches of the function.

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