Question

How do you determine if $5{{n}^{3}}+n{{q}^{3}}$ is a polynomial and if so, how do you identify if it is a monomial, binomial or trinomial?

Answer 1

Hint: First let us check if the given expression is a polynomial. A polynomial is an expression consisting of variables and the constants as their coefficients. The terms in a polynomial are bound together by any mathematical operations such as addition or subtraction or multiplication or division. To check if it is monomial or binomial or trinomial, count the number of terms.

Complete step by step answer:
The given expression is $5{{n}^{3}}+n{{q}^{3}}$
Let us first check if it is a polynomial.
Yes, it is a polynomial. It consists of three variables $n,q\;$ . The coefficient is $5$ for ${{n}^{3}}$ . The terms should also be bound together by any mathematical operations such as addition or subtraction or multiplication or division.
Here the terms are bound together by the mathematical operation addition.
Now coming to check if it is a monomial or binomial or trinomial,
The word polynomial when split,
“poly” means many, “nominal” means terms.
In the same way,
The word monomial when split,
“mono” means one, “nomial” means terms.
The word binomial when split,
“bi” means two, “nomial” means terms.
The word trinomial when split,
“tri” means three, “nomial” means terms.
Since the expression, $5{{n}^{3}}+n{{q}^{3}}$ has two terms,
Hence it is said to be a binomial.

Note:
This is a polynomial of a degree $3$ since there are two terms with power $3$(highest of all). It is also known as a three-degree polynomial equation. Since it is a degree $3$ polynomial, we shall have three roots as solutions when we solve it.