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?
$$f(x, y)=x y + y^2 + 2 x y^2 + y^3 – 3 x y^3 + x y^4,$$
Best Answer
As you mentioned,
Now the crucial point is that polynomials can be classified as monomial ( 1 term ) , binomial ( 2 terms ), trinomial (3 terms) , quadrinomial (4 terms), quintinomial (5 terms), multinomial ( polynomial having more than one terms ) etc depending on the number of terms present in their expressions.
So multinomial is a type of polynomial having more than one terms in it.
Therefore, $$f(x, y)=x y + y^2 + 2 x y^2 + y^3 - 3 x y^3 + x y^4,$$
is a multivariable multinomial polynomial i.e. it is a polynomial having more than one variable and more than one terms.