Multiplicity Question: Is answer "7" for (x-3)^4( x-4)(x-5)(x-8)^2 ?

markraz

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(x-3)^4( x-4)(x-5)(x-8)^2

Hi, I have a yes or no question with regards to the above
does multiplicity = 7?


thanks
 
Well, I've never seen the term multiplicity used in this particular context before, but it sounds like you're working with this definition from MathWords. Under this definition, the "multiplicity of a polynomial" would appear to be the sum of the multiplicities of its roots. So, what are the roots of your polynomial? (Hint: If (x - r) is a factor of a polynomial, then r is a root) What are the multiplicities of each root? (Hint: If (x-r)n is a factor of a polynomial, the root r has multiplicity n). What, then, would be the sum of all of these multiplicities? Is it 7?
 
(x-3)^4( x-4)(x-5)(x-8)^2

does multiplicity = 7?
That ought to read something like: Does this polynomial have 7 roots, counting multiplicity?

Hint: A polynomial of degree n has n roots, counting multiplicity.
 
(x-3)^4( x-4)(x-5)(x-8)^2

Multiplicity refers to one root only.

x = 3 with multiplicity 4
x = 8 with multiplicity 2
x = 4 with multiplicity 0 (or no multiplicity)

Multiplicity is also important in understanding vertical asymptotes of rational functions.

AS previously stated, we can talk of the DEGREE of the polynomial or the total number of roots, but adding "multiplicity" actually makes no sense. Since (x-4) has NO multiplicity, shall we count it as 1 or zero?
 
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Opinions vary and there is no significance known to me which way we refer to it.

I tend to use 0 for this reason.

For a cubic polynomial, (x-a)(x-b)(x-c), we usually refer to it having "three roots". We do not refer to multiplicity at all. Does it have any? Thus, all three might have multiplicity 0 or NO multiplicity.

For a cubic polynomial, (x-a)(x-a)(x-a), we usually refer to having "three roots, counting multiplicities". Are we counting all three or just the two extra?

I have no problem with other interpretations and I am unaware of an authoritative standard. I realize I'm trying to have to both ways, counting 0 for (x-a) and 3 for (x-a)^3. If I were particularly consistent, I would say 1 and 3 or 0 and 2. But, like I said, I don't find it all that significant. Personally, I want to be confusing so we don't rely on "multiplicity" as a well-defined term. I prefer to have the equivalent discussion referring to "degree".
 
\(\displaystyle (x - 3)^4(x - 4)(x - 5)(x - 8)^2 \ \ \) is a polynomial that has four distinct zeros.

The zero 3 has multiplicity 4, because it occurs four times.


The zero 4 has multiplicity 1, because it occurs one time.
It can never be said to have multiplicity 0, because it does occur.



The zero 5 has multiplicity 1, because it occurs one time.
It can never be said to have multiplicity 0, because it does occur.



The zero 8 has multiplicity 2, because it occurs two times.


The polynomial has 8 zeros, counting multiplicity.
 
Yes, that is one way to interpret it.

English Language Argument: "Multiplicity" necessarily refers to a plurality, not to a unity. "multiplicity of 1" is not a meaningful expression. A single occurrence has "no multiplicity". If you would like to count that as "1", feel free. Just be very clear what you want your students to put on an exam.

Arithmetic Argument: The polynomial also has four zeros, NOT counting multiplicities. (You called them "distinct zeros"). This leaves only four more things to count to get to the total of eight. Can you point out the extra 4? You will NOT be recounting those with Degree 1.

Like I said, opinions vary.
 
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Thanks peeps

Sorry for the confusion.

This is the reason I asked: go here https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-ex-1.html

please refer to at "Knots with Positive Multiplicity" section

Multiplic.jpg


I'm trying to understand what multiplicity means in this context above. In other words lets say hypothetically I added another value (.7) to this multi-set
{0, 0, 0, 0.3 0.5, 0.5, 0.6, 0.7, 1, 1, 1} .. would this still have a multiplicity of 9?
thanks
 
It does help to provide exactly what it is you are doing, rather than trying to ask a simpler question that is only barely applicable.

In any case, I'm sorry that I have not managed to discourage you from adding up "multiplicity". It doesn't work. Don't do it.

Here is a useful definition of a knot vector:

[FONT=&quot]"A knot vector can also be expressed as a vector of ascending distinct knot values and a vector of multiplicity for each of the distinct values."

Notice how the multiplicity applies to each knot value BY ITSELF. There is no mention of adding things up. "Multiplicity" applies to a single vector element. Each element has its own multiplicity. There is no definition of the multiplicity of the entire vector.

It is true that some texts and discussions (using a polynomial as an example) define "multiplicity" as "How many times does the factor or zero appear?" This is what lookagain defined, nicely. However, I always have found this definition confusing, for the two reasons I stated above. I prefer to refer to "degree" or "order". These are much more clearly and specifically defined.

One more time to emphasize ambiguity.

In lookagain's polynomial example, these two things can be said.

1) There are 8 zeros, counting multiplicities.
2) Ignoring multiplicities, we see 4 distinct zeros.

In #1, we count the linear factors as having multiplicity = 1. This gets us to 8 zeros. It does not get us to multiplicity = 8 for the entire polynomial. That still doesn't mean anything.
In #2, we are ignoring multiplicity, so we cannot be counting multiplicity = 1. There is no multiplicity. We're ignoring it! Yet, we still get to 4 distinct zeros. We must be counting something other than multiplicity.

Words mean things. We can't just play like we understand when we really don't. Terms must be clearly defined. I have yet to see a perfectly clear, indisputable, and universal definition of "multiplicity". This is why I prefer "order" or "degree". Their definitions are less disputable.

My views. I welcome others'.[/FONT]
 
I'm trying to understand what multiplicity means in this context above. In other words lets say hypothetically I added another value (.7) to this multi-set
{0, 0, 0, 0.3 0.5, 0.5, 0.6, 0.7, 1, 1, 1} .. would this still have a multiplicity of 9?
thanks

Before inserting the 0.7, we have:

{0, .3, .5, .6, 1} <== Distinct elements
{3, 1, 2, 1, 3} <== Multiplicities

After inserting the 0.7, we have:

{0, .3, .5, .6, 0.7, 1} <== Distinct elements
{3, 1, 2, 1, 1, 3} <== Multiplicities

Yes, this uses Multiplicity = 1 for single elements. Sometimes, that is what is wanted or needed. With this definition, you can add up the multiplicities and effectively count the total number of elements. The entire set, however, still doesn't have a multiplicity. It's only for distinct elements.
 
Yes, that is one way to interpret it.

English Language Argument: "Multiplicity" necessarily refers to a plurality, not to a unity. "multiplicity of 1" is not a
meaningful expression. A single occurrence has "no multiplicity". If you would like to count that as "1", feel free.
Just be very clear what you want your students to put on an exam.

No, it does not necessarily refer to a plurality. It refers to the number of multiples of it, whether that be 0, 1, or a
whole number more than 1. A good definition is given in the beginning of this PurpleMath page:

http://www.purplemath.com/modules/polyends2.htm


Like I said, opinions vary.

This isn't about opinions. This is about stating falsehoods and truths.

- - - - - - - -- - - -



It is true that some texts and discussions (using a polynomial as an example) define "multiplicity" as "How many times does the factor or zero appear?"
About those factors and zeros, it I clear, but you're going with a falsehood by claiming "multiplicity"to mean something else.

This is what lookagain defined, nicely. However, I always have found this definition confusing, for the two reasons I stated above. I prefer to refer to
"degree" or "order". These are much more clearly and specifically defined.

One more time to emphasize ambiguity.

In lookagain's polynomial example, these two things can be said.

1) There are 8 zeros, counting multiplicities.
2) Ignoring multiplicities, we see 4 distinct zeros.

In #1, we count the linear factors as having multiplicity = 1. This gets us to 8 zeros. It does not get us to multiplicity = 8 for the entire polynomial. That still doesn't mean anything.
In #2, we are ignoring multiplicity, so we cannot be counting multiplicity = 1. There is no multiplicity. We're ignoring it! Yet, we still get to 4 distinct zeros. We must be counting something other than multiplicity.


If multiplicity is being ignored, the number of occurrences of distinct factors is being ignored. Just the fact that the
factor is represented is what counts for the corresponding zero.

Is that a typo? I've always thought that simple roots have multiplicity 1.

"Now, we’ve got some terminology to get out of the way. If r is a zero of a polynomial and the exponent on the term that
produced the root is k then we say that r has multiplicity k. Zeroes with a multiplicity of 1 are often called simple zeroes."

http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx

https://en.wikipedia.org/wiki/Multiplicity_(mathematics)

And, mmm4444bot, you are correct about that.
 
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There is no "truth" in mathematics. There is validity and consistency within the framework of an axiomatic system.

[h=2]Definition of multiplicity - Merriam Webster[/h][FONT=&quot]plural multiplicities:the quality or state of being multiple or various
:a great number

:the number of times a root of an equation or zero of a function occurs when there is more than one root or zero
  • the multiplicity of x = 2 for the equation (x − 2)3 = 0 is 3


Emphasis added. Like I said in the first place, opinions vary.



[/FONT]
 
I was also taught (about terminology) that:

Polynomials have roots

Functions have zeros

Equations have solutions

Yet, even those who taught me this did not use these terms consistently, as they had defined them. (I had visited the math library at the University of Washington, and looked through some 50- to 70-year old textbooks; they were not consistent, either, saying things like, "Find the roots of this equation.")

Math has a lot of variability, both in the way it's taught and used:

"zero is either positive or negative"
versus
"zero is neither positive nor negative"

e^x = y implies that ln(y) = x
versus
e^x = y implies that log(y) = x

0^0 is undefined
versus
0^0 = 1

We all have to get used to variability. :cool:
 
:) I've just never been in a discussion where a technical distinction of "root" and "zero" made any difference. However, using either makes me a little nervous - considering I might be talking to someone who will be offended if I use the wrong one.
 
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There is no "truth" in mathematics.
There are truths (read: rigid facts) in portions of mathematics. There are variations and inconsistencies in the definitions of many mathematical terms.

mmm4444bot said:
Math has a lot of variability, both in the way it's taught and used:

"zero is either positive or negative"
versus
"zero is neither positive nor negative"

e^x = y implies that ln(y) = x
versus
e^x = y implies that log(y) = x

0^0 is undefined
versus
0^0 = 1

Second example: Those logarithm words are different, but they are still are used for the same base.

Third example: 0^0 is undefined. For instance, there is a "hole" on the graph of y = x^x. However,
for the graph of y = x^x, the limit of x^x, as x approaches 0 from the right, is equal to 1.
 
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There are truths (read: rigid facts) in portions of mathematics.

Nope. Change an axiom, get a different system.

There are variations and inconsistencies in the definitions of many mathematical terms.

Can't argue with that. The ubiquitous \(\displaystyle \pm\) is an excellent example. Can you name five different ways it is used?

Second example: Those logarithm words are different, but they are still are used for the same base.

"log" can change. When I first encountered logarithms, "log" meant Base 10 only. Later, the appearance of "log" meant Base e. Never assume.

Third example: 0^0 is undefined.

Maybe, Depends on what you mean by the notation.
 
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