Question

Which of the following expressions are polynomial in one variable and which are not? State reason for your answer.
(a) $4{{x}^{2}}-3x+7$
(b) ${{y}^{2}}+\sqrt{2}$
(c) $3\sqrt{t}+t\sqrt{2}$
(d) $y+\dfrac{2}{y}$
(e) ${{x}^{10}}+{{y}^{3}}+{{t}^{50}}$

Answer 1

Hint:In a given question, we will use the concept of variable in algebraic expression to find the number of variables in the given question. For a polynomial to be of one variable, there should be only one unknown quantity in the polynomial.

Complete step-by-step answer:
In algebra, variable is an unknown number which is represented by symbols such as letters. Value of this variable depends on the conditions associated with the algebraic expression in which it is used. It is known as variable because its value in the expression is not fixed and can vary depending on equality of expression.
Also, an expression in algebra can contain any number of variables from zero to infinite.
Now, if in a polynomial, only one variable is used, we call it polynomial in one variable.
If two variables are used, we call it polynomial in two variables and so on.
Now, the first expression given to us in the question is, $4{{x}^{2}}-3x+7$. Here we can see that, only one letter is used and rest are fixed numbers. So, this is polynomial in one variable.
The second expression given to us in question is ${{y}^{2}}+\sqrt{2}$. Here we can see that only one letter, which is variable y is used and the rest is a fixed number.
So, this is also a polynomial in one variable.
The third expression given to us in a question is $3\sqrt{t}+t\sqrt{2}$. Here also, only one letter is used, which is variable t and rest are fixed numbers.
So, this is also a polynomial in one variable.
The fourth expression given to us in a question is $y+\dfrac{2}{y}$. Here also, one letter, that is variable y is used and rest is fixed numbers.
So, this is also polynomial in one variable.
The fifth expression given to us in a question is ${{x}^{10}}+{{y}^{3}}+{{t}^{50}}$. Here, we can see three different letters, which are variables x, y and t are used.
So, this is polynomial in three variables.
Hence, $4{{x}^{2}}-3x+7$, ${{y}^{2}}+\sqrt{2}$, $3\sqrt{t}+t\sqrt{2}$ and $y+\dfrac{2}{y}$ are polynomials in one variable.

Note: In this type of question, when we are to find variables of polynomials, note that power of variable on the function in which variable is used does not matter. and are the same variable with different powers. Students generally get confused and treat them as two variables, which is wrong.