Question

Which of the following expressions are polynomials in one variable and which are not ? State reasons for your answer:

(i) $3x^2-4x+15$

(ii) $y^2+2\sqrt{3}$

(iii) $3\sqrt{x}+\sqrt{2}x$

(iv) $x-\frac{4}{x}$

(v) $x^{12}+y^3+t^{50}$

Answer 1

For an algebraic expression to be a polynomial, the power of the variables should be a whole number.

For an algebraic expression to be a polynomial in one variable, the variable should be of only one type whose power is a whole number.

Now, for,

(i) $3x^2-4x+15$ - The variable used is of one type only which is $x$ whose powers are whole number. Hence, the given expression is a polynomial in one variable.

(ii) $y^2+2\sqrt{3}$ - The variable used is of one type only which is $y$ whose power is a whole number. Hence, the given expression is a polynomial in one variable.

(iii) $3\sqrt{x}+\sqrt{2}x$ - The variable used is of one type only which is $x$ but the power of the variable is not a whole number, rather it is a fraction as $\sqrt{x}=x^{\frac{1}{2}}$. Hence, the given expression is not a polynomial.

(iv) $x-\frac{4}{x}$ - The variable used is of one type only which is $x$ but the power of the variable is not a whole number, rather it is a fraction or an integer as $\frac{4}{x}=4x^{-1}$. Here the power of $x$ is $-1$ which is not an integer. Hence, the given expression is not a polynomial.

(v) $x^{12}+y^3+t^{50}$ - The variables used in the given expression are more than one type ie . $x,y$ and $z$. Also, the powers of the respective variables is a whole number. Hence, the given expression is a polynomial but not one variable.

Hence, (i) and (ii) are polynomials in one variable.