Question

Which of the following expressions are polynomials in one variable and which are not? State reasons 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.${x^{10}} + {y^3} + {t^{30}}$

Answer 1

Hint: A polynomial is an algebraic expression which consists of variables and coefficients. A polynomial in one variable is that polynomial which consists of one and only one variable irrespective of the number of times it is repeated.

Complete step-by-step answer:
The given polynomial expression is $4{x^2} - 3x + 7$.
Since it is a polynomial with only one variable i.e. $x$. Thus it is a polynomial in one variable.
The given polynomial expression is ${y^2} + \sqrt 2 $
Since it is a polynomial with only one variable i.e. $y$. Thus it is a polynomial in one variable.
The given polynomial expression is $3\sqrt t + t\sqrt 2 $
Since it is a polynomial with only one variable i.e. $t$. Thus it is a polynomial in one variable.
The given polynomial expression is $y + \dfrac{2}{y}$
For any polynomial expression, division by variable is not allowed. Thus it is not a polynomial expression.
The given polynomial expression is ${x^{10}} + {y^3} + {t^{30}}$
Since it is a polynomial with three variables i.e. $x,y$ and $t$ . Thus it is not polynomial in one variable.

Note: A polynomial is classified on the basis of 2 types:
1.Order of the polynomial
2.Number of variables.
Order of a polynomial is defined as the highest power of any variable in the polynomial. For example, ${x^{10}} + {y^3} + {t^{30}}$.. In this expression, the highest power of a variable is 30 so the order of the variable is 30.