Question

Which of the following expressions are polynomials in one variable and which are not? Give reason for your answer.
a) \[3{{x}^{2}}-\text{ }4x\text{ }+\text{ }15\]
b) \[{{y}^{2}}+\text{ }2\sqrt{3}\]
c) \[3~\sqrt{x}+~\sqrt{2}x\]
d) \[x-\dfrac{4}{x}\]
e) \[{{x}^{12}}+\text{ }{{y}^{3}}+\text{ }{{t}^{50}}\]

Answer 1

Hint: Before solving this question, we must know about polynomials and polynomials in one variable.
Polynomials: Polynomials are algebraic expressions that comprise of exponents which are added, subtracted or multiplied. Polynomials are of different types: namely Monomial, Binomial, and Trinomial. A monomial is a polynomial with one term. A binomial is a polynomial with two, unlike terms.
Polynomial in one variable: When there is only a single variable in the polynomial expression, then that polynomial is called a polynomial in one variable. We generally denote that single variable by x (or in some cases, y or z).

Complete step-by-step answer:
Let us now solve this question.
We shall consider every option.
a)\[3{{x}^{2}}-\text{ }4x\text{ }+\text{ }15\]
We can see that in this polynomial, there is only one variable, i.e. ‘x’
 Therefore, this is a polynomial in one variable.

b)\[{{y}^{2}}+\text{ }2\sqrt{3}\]
We can see that in this polynomial, there is only one variable, i.e. ‘x’
Therefore, this is a polynomial in one variable.

c)\[3~\sqrt{x}+~\sqrt{2}x\]
We can see that in this polynomial, there is only one variable, i.e. ‘x’
Therefore, this is a polynomial in one variable.

d)\[x-\dfrac{4}{x}\]
We can see that in this polynomial, there is only one variable, i.e. ‘x’
Therefore, this is a polynomial in one variable.

e)\[{{x}^{12}}+\text{ }{{y}^{3}}+\text{ }{{t}^{50}}\]
We can see that in this polynomial, there are three variables, i.e. ‘x’, ‘y’ and ‘t’
Therefore, this is not a polynomial in one variable.
Hence, the answers of this question are (a), (b), (c), and (d).

Note: Let us now learn about monomials, binomials, trinomials and terms.
MONOMIALS: A monomial is a polynomial with one term. For example: \[2xy,\text{ }3{{a}^{3}}\] , etc.
BINOMIALS: A binomial is a polynomial with two, unlike terms. For example: \[2xy\text{ }+\text{ }3{{x}^{2}},\text{ }3{{a}^{3}}-\text{ }5y\], etc.
TRINOMIALS: A trinomial is a polynomial with three terms, which are unlike. For example:
\[~2xy\text{ }+\text{ }3{{x}^{2}}+\text{ }4,\text{ }3{{a}^{3}}-\text{ }5y\text{ }+\text{ }8\] , etc.
TERMS: A term is either a single number or variable, or the product of several numbers or variables. Terms are separated by a + or - sign in an overall expression. For example: In the trinomial \[2xy\text{ }+\text{ }3{{x}^{2}}+\text{ }4;\text{ }2xy,\text{ }3{{x}^{2}}\] , and 4 are the three separate terms.