What is order of multiplicity?

What Does order of multiplicity Mean

Before entering fully into the meaning of the term order of multiplicity, we are going to proceed to know the etymological origin of the two main words that give it its shape:

-Order, first of all, it derives from Latin. Specifically, it emanates from “ordo, ordinis”, which can be translated as “order”.

-Multiplicity, secondly, comes from the Latin word “multiplicitas”, which is equivalent to “quality of having many ways”. It is a word that is the result of the sum of the following lexical components: the prefix "multus-", which is synonymous with "many"; the verb "plicare", which can be translated as "make folds"; and the suffix “-dad”, which is used to indicate “quality”.

If we want to define precisely the idea of order of multiplicity , it is necessary first to review several terms from the field of mathematics . Otherwise, understanding the expression will be very difficult.

In this framework, it is worth referring to the concept of a multiset . This is the name of the set in which each member is linked to a multiplicity that indicates how many times the element in question is a member of the set .
In the multiset {a, a, a, a, b, c} , for example , the multiplicity of a is 4 , while the multiplicity of b and c is 1 .
On the other hand, it is important to bear in mind that polynomials are expressions formed by at least two algebraic terms that are joined by a minus sign ( - ) or by a plus sign ( + ). Finally, the notion of root must be considered as the value that, in an equation, the unknown can have.
The root of a polynomial, then, is a number that allows the polynomial to be annulled: when finding the numerical value, the result of the polynomial is 0 .
Now yes, we can move forward and focus on what is the order of multiplicity . It is the number of times a root is repeated in a polynomial . To determine it, it is necessary to factor the polynomial.
In other words , the order of multiplicity refers to how many times a certain number is the root of a polynomial . For example, if the root of a polynomial is 4 , the number of times that 4 appears as the root of that polynomial will be its order of multiplicity.
In the same way, it is necessary to know that the order of multiplicity becomes a very important element to be able to determine what is the behavior of a polynomial function with respect to what is the so-called "X" axis, which is also known as the abscissa axis.

In addition to all the above, there is another series of questions related to the order of multiplicity that is worth taking into account:

-If the aforementioned order of a root turns out to be even, the graph of the function in question touches the so-called "X" axis , but it does not go through. That is, it bounces.

-In the event that the order of multiplicity of a root is odd, the graph of this function crosses the aforementioned "X" axis, that is, it cuts it.

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