Which one is polynomial?

Options

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

Explanation

To determine which of the given expressions is a polynomial, we need to analyze each option based on the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and combined using addition or subtraction.

### Step-by-Step Solution:

1. **Identify the Expressions:**
- Option A: \(1 + \sqrt{x}\)
- Option B: \(x^2 + 1\)
- Option C: \(\frac{1}{x^3} + x^2 + x\)
- Option D: \(\frac{1}{x^3}\)

2. **Analyze Option A: \(1 + \sqrt{x}\)**
- The term \(\sqrt{x}\) can be rewritten as \(x^{1/2}\).
- The exponent \(1/2\) is not a non-negative integer (it is a fraction).
- Therefore, Option A is **not a polynomial**.

**Hint:** Check if all exponents in the expression are non-negative integers.

3. **Analyze Option B: \(x^2 + 1\)**
- The term \(x^2\) has an exponent of 2, which is a non-negative integer.
- The constant term 1 can be considered as \(1 \cdot x^0\), where the exponent is 0 (also a non-negative integer).
- Therefore, Option B is a **polynomial**.

**Hint:** Look for terms with non-negative integer exponents.

4. **Analyze Option C: \(\frac{1}{x^3} + x^2 + x\)**
- The term \(\frac{1}{x^3}\) can be rewritten as \(x^{-3}\).
- The exponent \(-3\) is a negative integer, which violates the polynomial condition.
- Therefore, Option C is **not a polynomial**.

**Hint:** Ensure that all terms have non-negative integer exponents.

5. **Analyze Option D: \(\frac{1}{x^3}\)**
- Similar to Option C, this can be rewritten as \(x^{-3}\).
- Again, the exponent is \(-3\), which is negative.
- Therefore, Option D is **not a polynomial**.

**Hint:** Remember that a polynomial cannot have negative exponents.

### Conclusion:
The only expression that qualifies as a polynomial is **Option B: \(x^2 + 1\)**.