Which one is polynomial?

Options

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

Explanation

To determine which of the given expressions is a polynomial, we need to review the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and has coefficients that are real numbers. Let's analyze each option step by step.

### Step-by-Step Solution:

1. **Option A: \( \sqrt{x} + 1 \)**
- The expression \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).
- Since \( 1/2 \) is not a non-negative integer, this expression is **not a polynomial**.

2. **Option B: \( \frac{x}{x^3 + 1} \)**
- This expression is a fraction. The denominator \( x^3 + 1 \) is a polynomial, but the entire expression \( \frac{x}{x^3 + 1} \) is not a polynomial because it involves division by a polynomial, which can lead to undefined values (especially when \( x^3 + 1 = 0 \)).
- Therefore, this expression is **not a polynomial**.

3. **Option C: \( \frac{1}{x^2 + 1} \)**
- Similar to option B, this expression is also a fraction. The denominator \( x^2 + 1 \) is a polynomial, but the entire expression \( \frac{1}{x^2 + 1} \) is not a polynomial due to the division.
- Thus, this expression is **not a polynomial**.

4. **Option D: \( x^3 + 1 \)**
- This expression consists of \( x^3 \) (which is a non-negative integer power) and a constant term \( 1 \).
- Since both terms have non-negative integer exponents, this expression satisfies the definition of a polynomial.
- Therefore, this expression **is a polynomial**.

### Conclusion:
The only expression that qualifies as a polynomial from the given options is **Option D: \( x^3 + 1 \)**.