Question

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

(i) $3\sqrt{t}+t\sqrt{2}$

(ii) $y+\frac{2}{y}$

(iii) $x^{10}+y^3+t^{50}$

Answer 1

Step 1: (i) Determine whether the expression $3\sqrt{t}+t\sqrt{2}$ is polynomial in one variable or not.

A polynomial is defined as a mathematical equation with one or more terms made up of variables, constants, and exponents that are mixed using operations like addition, subtraction, multiplication, and division.

The expression $3\sqrt{t}+t\sqrt{2}$ can be written as follows:

$3\sqrt{t}+t\sqrt{2}=3{\left(t\right)}^{\frac{1}{2}}+t{\left(2\right)}^{\frac{1}{2}}$

In the given expression, $t$ is the variable that has power $\frac{1}{2}$, which is not a whole number.

This implies that the given expression is not a polynomial.

Hence, $3\sqrt{t}+t\sqrt{2}$ is not a polynomial in one variable.

Step 2: (ii): Determine whether the expression $y+\frac{2}{y}$ is polynomial in one variable or not.

The expression $y+\frac{2}{y}$ can be written as follows:

$y+\frac{2}{y}=y+2y^{-1}$

In the given expression, $y$ is the variable that has power $-1$, which is not a whole number.

This implies that the given expression is not a polynomial.

Hence, $y+\frac{2}{y}$ is not a polynomial in one variable.

Step 3: (iii): Determine whether the expression $x^{10}+y^3+t^{50}$ is polynomial in one variable or not.

In the given expression, $x,y,z$ are the variables that have power $10,3,50$, which is a whole number.

This implies that the given expression is a polynomial but not a one-variable polynomial.

Hence, $x^{10}+y^3+t^{50}$ is not a polynomial in one variable.