Math question about zeros of polynomial functions?

I am currently studying about Zeros of polynomial functions. While I understand how to do everything to the letter in this particular segment, I am having trouble understanding what the purpose of a certain aspect is, I am not at all discounting that it is important. I am truly interested. Here goes: ... show more I am currently studying about Zeros of polynomial functions. While I understand how to do everything to the letter in this particular segment, I am having trouble understanding what the purpose of a certain aspect is, I am not at all discounting that it is important. I am truly interested.

Here goes: "Definition of multiple zeros of a polynomial function." If a polynomial function P has (x-r) as a factor exactly k times, then r is a zero of multiplicity k of the polynomial function P.

Why does this seem as if it is an arbitrary statement? The outcome of the function does not change so why is it important to become aware of the inputs? I want to know where I am going to see this later on so I can recognize it better then as well as understand it 100%.

An example of a problem:

P(x) = (x-5)^2(x+2)^3(x+4)
(statements)
*5 as a Zero of multiplicity 2
*-2 as a Zero of multiplicity 3
*-4 as a Zero of multiplicity 1
Update: So once again, my question is: What is the value of knowing the "multiplicity" and where in higher math can I expect to see this used? I am purely interested in the use of these unusual ideas. I asked my Prof. and he equivocated his answer. I am afraid he was more concerned with the rest of the... show more So once again, my question is: What is the value of knowing the "multiplicity" and where in higher math can I expect to see this used? I am purely interested in the use of these unusual ideas.

I asked my Prof. and he equivocated his answer. I am afraid he was more concerned with the rest of the class who is suffering with the chapter in general.

Any help will be very appreaciated!
Thanks!
5 answers 5