Q1(iv):

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$

Step 1: Initial Setup & Algebraic Transformation

The given algebraic expression is:

$y + \frac{2}{y}$

To rigorously analyze whether this expression is a polynomial, we must first express all terms such that the variable appears in the numerator. [Per the Laws of Exponents, specifically the negative exponent rule $\frac{1}{a^n} = a^{-n}$], we rewrite the second term of the expression:

$y^1 + 2y^{-1}$

Step 2: Variable Analysis

The problem requires us to verify two distinct conditions: whether the expression is in one variable, and whether it is a polynomial.

Looking at the transformed expression $y^1 + 2y^{-1}$, the only alphabetical symbol representing an unknown quantity is $y$. There are no other variables (such as $x$, $z$, etc.) present. Therefore, the expression strictly satisfies the condition of being in one variable.

Step 3: Theoretical Definition of a Polynomial

By mathematical definition, an algebraic expression is classified as a polynomial if and only if it can be written in the form:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

Where:

• $a_n, a_{n-1}, \dots, a_0$ are real number coefficients.
• The exponents $n, n-1, \dots$ must belong strictly to the set of non-negative integers (i.e., the set of whole numbers, $\mathbb{W} = \{0, 1, 2, 3, \dots\}$).

Step 4: Exponent Analysis

We now evaluate the exponents of the variable $y$ in our transformed expression $y^1 + 2y^{-1}$ against the theoretical definition:

• First Term ($y^1$): The exponent of $y$ is $1$. Because $1 \in \mathbb{W}$, this specific term satisfies the polynomial condition.
• Second Term ($2y^{-1}$): The exponent of $y$ is $-1$. Because $-1 \notin \mathbb{W}$ (it is a negative integer, not a whole number), this term violates the fundamental definition of a polynomial.

Step 5: Logical Conclusion

For an entire expression to be classified as a polynomial, every single term must possess a whole number exponent. The presence of even one term with a negative or fractional exponent immediately disqualifies the entire expression.

Final Solution: The expression $y + \frac{2}{y}$ is NOT a polynomial. While it is an expression in exactly one variable ($y$), the exponent of the variable in the second term is $-1$, which is not a whole number.