Question

Which of the following expressions are polynomials in one variable and which are not?

State reasons for your answer. (i) $y^2+\sqrt{2}$ (ii) $3\sqrt{t}+t\sqrt{2}$ and (iii) $x^{10}+y^3+t^{50}$.

Answer 1

Step 1: (i): Determine whether the expression $y^2+\sqrt{2}$ is a 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 $y^2+\sqrt{2}$ can be written as follows:

$y^2+\sqrt{2}=y^2+\sqrt{2}{y}^0$

In the given expression, $y$ is the variable that has powers $2\mathrm{and}0$, which are whole numbers.

Hence, $y^2+\sqrt{2}$ is a polynomial in one variable.

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

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

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

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

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

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

In the given expression, $x,y,t$ are the variables that have powers $10,3,\mathrm{and}50$, which are whole numbers.

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