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: "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)
*5 as a Zero of multiplicity 2
*-2 as a Zero of multiplicity 3
*-4 as a Zero of multiplicity 1
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!
- 1 decade agoBest Answer
You are right that the definition seems a bit oblivious, why make it?
There are a couple of places where knowing the multiplicity of a root may help you out. And if you know the multiplicity of a root then you may be able to find other roots and completely factor the polynomial. For instance if you know the complete factorization of a polynomial then you can pretty much graph the thing without a graphing calculator, someone has already mentioned:
If a polynomial f(x) has a root (or zero) multiplicity of m at x=r then the graph of the function will touch the x-axis at x=r if m is even; and cross the x-axis at x=r if m is odd.
Knowing the multiplicity of a single root can help one find all the zeros of a polynomial, which is not an easy task.
At any rate you are correct that the input or the output in not changing... what is the shape of the graph.... you may want to compare x, x^2, x^3, x^4.... do you see how the shape is changing, however for all of these we have that when x=0 that y=0. So we see that the inputs and the outputs are the same so what's the difference? The degree of the function, and hence the degree of the roots.
The first thing that one should do when you find a single zero is to see if that zero is a multiple root, then the left over polynomial is a new polynomial that you can begin anew with. In this way you can get a complete factorization of your polynomial.
Perhaps one might want to tell a polynomial apart from another by it's roots, if you did you would need to know the degree of each of the roots of the polynomial.
Indeed, someone else has also mentioned the Fundamental Theorem of Algebra (but not actually stated it) This is probably going to be where you are trying to get to in this chapter of your class. But basically is says that:
Given a polynomial f(x), f(x) can be completely factored as
where z is a complex number and that a polynomial of degree n has n complex roots (up to multiplicity of n).
As I said people above me have hinted at all this, so what can I add to the discussion rather then rehashing what they said to make things clearer.
Let me attempt to put this it's larger context, let me remind you about PRIME factorization of a number, right for an integer we can decompose it into it's prime factors
EXs 6 = 2*3; 8=2^3; 440=2^3*5*11
Once you know the prime factors you know everything important about the number.
Compare prime decomposition with your example
P(x) = (x-5)^2(x+2)^3(x+4).
Notice any similarities? (hopefully so)
Here what are your "prime factors"? (x-5) ; (x+2); (x+4)
Suppose you wanted to `abstract' the prime decomposition idea to polynomials... indeed this is exactly what the multiplicity of a root is, basically telling you haw many times a "prime factor" of a smaller polynomial goes into a polynomial. In fact in abstract algebra this is pretty much what it means for a polynomial to be "prime". ( This is a HUGE over simplification ). As if this wasn't enough of a reason right there on why this might be interesting, I can't help but mention Galois Theory.
Why is this useful? Well as I already said it is very hard to find roots of polynomial equations. 100s of years ago prizes were being offered to people that could find formulas (like the quadratic formula) to find all roots to polynomial equations. They were stuck on a degree 5 polynomial equations. That is They wanted to find all roots to a polynomial
0 = ax^5 + bx^4 +cx^3 +dx^2 +cx+e
Then a guy named Galois (about 17 years old) comes up and says "it's not possible to find formulas for degrees 5 and higher." He turned out be be correct, and consequently did not get the prize because he didn't find a formula.
However, this still shocks me to this day, you may want to read up on Galois Theory:
What was the point of that story? In essence a part of this 'Galois Theory' deals with these "prime polynomials". Of course he developed A LOT of other mathematics (I mean he has an Theory named after him which is a series of statements, I think he came up with the name 'group' which is a very common math term) to fully understand the profoundness of what he did will take someone a good couple of years in Abstract Algebra.
This whole chapter that you are studying about relations between roots, and zeros and polynomials has always fascinated me. There is quite a bit of interesting theory going on in the background here that you may never be able to see.
When you learn math, you learn it from the bottom up leaving one to ask "what's the point?", "why define this?" However when you look at it from the top down you suddenly realize "AH WOW! That's actually kind of amazing."
I hope this very long and over complicated answer gives you the insight that you were looking for.Source(s): Myself.
- Anonymous1 decade ago
I do not understand what you mean the "OUTCOME" of the function does not change & aware of the "INPUTS". No idea what you are trying to ask.
Lets say we have P = x² + 41x +141 and I tell you I have found a root of that, say its "Z". Since it is a quadratic, we know that there are two roots. Would you ask me for the other root or can we somehow determine that the two roots are degenerate and we can stop looking for more? It would be nice to know. I'm sure there are other reasons to differentiate between degenerate polynomial roots and non-degenerate ones, but thats all I got. You are right, it is a definition (and so arbitrary in that sense) not a proof. But I always say, once you understand a discipline's terminology you understand 80% of the discipline. I seem to recollect some algorithms for solving for polynomial roots may run into trouble with multiple roots. Not sure if state-of-the-art algorithms have that problem. I'm talking where the Order of P is huge.
- j gLv 51 decade ago
Knowing multiplicity is important because it will let you know whether the function will cross the x-axis at that point or just touch it. In general, a zero of odd multiplicity will cross the x-axis while one of even multiplicity will only touch it.
This will help you to be able to see the behavior of the curve at certain values, allowing you to sketch them. You will use this information to see what the behavior will be, such as whether the graph crosses the x-axis and continues going up or down, or if the graph touches the x -axis and then turns back up or turns down.
Taking your example, with 5 as a zero of multiplicity 2, this is an even multiplicity so the graph will only touch the x-axis and turn back at x = 5.
-2 as a zero of multiplicity 3, this is an odd multiplicity so the graph will cross the x-axis at x =-2.
With -4 of even multiplicity, the graph will only touch the x-axis at x = -4.
I hope this helps.
- William BLv 71 decade ago
I am not sure I am answering you question but:
the Fundamental Theorem of Algebra says that a polynomial equation x^n ...... =0 has n roots.
If above you said that P(x) has roots 5,-2, -4 if would be incomplete and it might seem that it is a third degree equation, instead of a 6th degree. Expressing multiplicities is a way of being in sync with the Fundamental Theorem.
Not sure what you mean by inputs and outputs. I don't think of factors as inputs.
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- 1 decade ago
I think you probably already know what I'm going to say, but one helpful part is that an even multiplicity will touch the x-axis and not cross it, but an odd one will cross it.
Also, arbitrary means chosen without reason (sort of like random but not completely). I think the world you're looking for is trivial which is the opposite of important.