Which one is polynomial?

Options

A. `(1)/(x+1)`
B. `x^((1)/(3))+2`
C. `(1)/(x^(2)+1)`
D. `x+sqrt2`.

Explanation

To determine which of the given options is a polynomial, we need to recall the definition of a polynomial. A polynomial is an algebraic expression that meets the following criteria:

1. It consists of variables and coefficients.
2. The coefficients are real numbers.
3. The variables are raised to non-negative integer powers (0, 1, 2, ...).
4. The operations involved are addition, subtraction, and multiplication. Division by a variable is not allowed.

Now, let's analyze each option:

**Option 1: \( \frac{1}{x} + 1 \)**

- This expression involves division by the variable \( x \).
- Since division by a variable is not allowed in polynomials, this option is **not a polynomial**.

**Option 2: \( x^{\frac{1}{3}} + 2 \)**

- Here, the variable \( x \) is raised to the power of \( \frac{1}{3} \).
- The exponent \( \frac{1}{3} \) is not a non-negative integer (it is a fraction), so this option is **not a polynomial**.

**Option 3: \( \frac{1}{x^2} + 1 \)**

- Similar to the first option, this expression also involves division by the variable \( x^2 \).
- Thus, this option is also **not a polynomial**.

**Option 4: \( x + \sqrt{2} \)**

- In this expression, \( x \) is raised to the power of 1 (which is a non-negative integer).
- \( \sqrt{2} \) is a constant and does not involve any variables.
- Since both terms meet the polynomial criteria, this option is a **polynomial**.

**Conclusion:**
The only polynomial among the options given is **Option 4: \( x + \sqrt{2} \)**.