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What is the difference between "Polynomial" and "Multinomial" in two or more variables?

Since, by definition:

Multinomial:

An algebraic expression having two or more (unlike) terms is called a Multinomial.

For example:

$5x^2 - 2x$ is a multinomial having $2$ terms,

$5x^3- 2xy + 7y^2$ is a multinomial having $3$ terms,

$7xy - 9yz + 6zx - 7$ is a multinomial having $4$ terms.

Polynomials in two or more variables:

An algebraic expression in two or more variables is called a Polynomial if the Power of every variable in each term is a whole number.

Some books say "Multinomial" is one of the types of "Polynomial", and the other discuss it in particular.

Is the function $f$ cross "Polynomial" or "Multinomial"? Why?

$$f(x, y)=x y + y^2 + 2 x y^2 + y^3 - 3 x y^3 + x y^4,$$

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    $\begingroup$ A multinomial is simply a polynomial which is not a monomial. So, for example, your $f(x,y)$ is both a polynomial and a multinomial. A polynomial which is not a multinomial is a monomial, e.g. $3x^2$ or $4xyz^5$. $\endgroup$ – Crostul Apr 9 '17 at 7:24
  • $\begingroup$ @Crostul Is multinomial considered a type of polynomial $\endgroup$ – Emad kareem Apr 9 '17 at 7:26
  • $\begingroup$ Yes. All multinomials are polynomials. $\endgroup$ – Crostul Apr 9 '17 at 8:24
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    $\begingroup$ Multinomial is not a common word in mathematics and I think it's not worth the effort to find its meaning. Depending in the context I would take it as a synonymous of polynomial in several variables or monomial in several variables. $\endgroup$ – nowhere dense Apr 10 '17 at 16:33
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    $\begingroup$ It's likely to be a dated word that is still in use in some situations that deal with old mathematics. For example it's used in the "multinomial theorem" or in the "multinomial distribution". I am not totally sure, though. $\endgroup$ – nowhere dense Apr 10 '17 at 18:27
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The poly- of polynomial and the multi- of multinomial are, respectively, Greek and Latin prefixes meaning the same: many; and the -nomial is a Latin suffix meaning name. So both polynomial and multinomial just mean "many names". It is the interpretation of what "many names" is, precisely, and I don't think that there is a precise difference made that is agreed universally - you interpret by the context.

If you used the term binomial you'd expect a term such as $(x+y)$ in which there are two (bi-) names: $x$ and $y$; but you would not disagree that the term ${x^2+2x+1}$ was referred to as a polynomial but has only one name $x$, but this expression should be referred to as a monomial, but that term is rarely used.

My considered explanation for the inconsistency and confusion is in the interpretation of the suffix, -nomial, in which it should refer to name ($x$, $y$, ...) but some have interpreted it as referring to number (the indices).

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Under many definitions, polynomials and multinomials are synonyms (as commented above). Under any definition, your $f(x,y)$ is both a polynomial and a multinomial. A monomial is not a polynomial nor a multinomial. Poly $\to$ many, multi $\to$ multiple; neither multiple nor many can refer to one. Polynomial is used more commonly though my experience is that multinomial is more commonly used when referring to plotted distributions. Whereas in describing a function polynomial is used almost exclusively.

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    $\begingroup$ A monomial IS a polynomial. $\endgroup$ – Alex M. Mar 24 '18 at 14:08
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A multinomial is not a type of polynomial. Polynomials are limited to positive integer powers of variables. A multinomial can contain, say, square roots of a variable. Or other irrational functions of variables.

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  • $\begingroup$ can you give references for that? Because i never heard it ^^" $\endgroup$ – JayTuma Aug 18 '18 at 22:02
  • $\begingroup$ I just happened to be working through "Shaum's Outline of College Algebra" ed 4. A polynomial is monomial or multinomial in which every term is integral and rational. Spiegel gives lots of examples of multinomials and polynomials. "integral" means positive integer powers. multinomials are very different animals. $\endgroup$ – huckfinn Aug 18 '18 at 22:17
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Multinomials are generally distinct from polynomials. You can't have a $1/x$ term in a polynomial, for instance. It may be that those terms have become synonyms through common misusage.

I did see something interesting in a short little book POLYNOMIAL EQUATIONS: SYSTEMATIC THEORY SUMMARY, CHALLENGING EXAMPLES AND PROBLEMS.

A multinomial can often be converted to a polynomial by a simple variable substitution.
So if you have a multinomial of $1/x$, you substitute $z= 1/x$, then you have a polynomial in the variable $z$. Of course the new equation is then not valid at the singularity.

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