Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4 x^{2} - 3 x + 7$
(ii) $y^{2} + \sqrt{2}$

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Solution

Step 1: (i) Determine whether the expression $4 x^{2} - 3 x + 7$ 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 $4 x^{2} - 3 x + 7$ can be written as follows:
$4 x^{2} - 3 x + 7 = 4 x^{2} - 3 x^{1} + 7 x^{0}$
In the given expression, $x$ is the variable that has powers $2, 1, and 0$ , which are whole numbers.
Hence, $4 x^{2} - 3 x + 7$ is a polynomial in one variable.
Step 2: (ii): Determine whether the expression $y^{2} + \sqrt{2}$ is polynomial in one variable or not.
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 $2 and 0$ , which are whole numbers.
Hence, $y^{2} + \sqrt{2}$ is a polynomial in one variable.

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