Which of the following is a polynomial function ?

Options

A. `(x ^(2) -1)/( x ), x ne 0`
B. ` x ^(2) + 3x ^(2) -4x + sqrt2 x ^(-2)`
C. `(3x ^(2) + 7x -1)/(3)`
D. `2x^(2) + sqrtx +1`

Answer

To determine which of the given options is a polynomial function, we need to understand the definition of a polynomial function. A polynomial function is expressed in the form:

\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_1 x + a_0 \]

where:
- \( n \) is a non-negative integer (0, 1, 2, ...),
- \( a_n, a_{n-1}, ..., a_0 \) are constants (coefficients),
- The powers of \( x \) must be whole numbers (non-negative integers).

Now, let's analyze each option step by step:

### Step 1: Analyze Option 1
**Option 1: \( x^2 - \frac{1}{x} \)**

- Rewrite: \( x^2 - x^{-1} \)
- The term \( x^{-1} \) has a negative exponent, which violates the definition of a polynomial function.

**Conclusion:** This is NOT a polynomial function.

### Step 2: Analyze Option 2
**Option 2: \( \frac{1}{x^2} \)**

- Rewrite: \( x^{-2} \)
- The term \( x^{-2} \) has a negative exponent, which also violates the definition of a polynomial function.

**Conclusion:** This is NOT a polynomial function.

### Step 3: Analyze Option 3
**Option 3: \( 3x^2 + 2x + 1 \)**

- The terms are \( 3x^2, 2x, \) and \( 1 \).
- All the powers of \( x \) are non-negative integers (2, 1, and 0).

**Conclusion:** This IS a polynomial function.

### Step 4: Analyze Option 4
**Option 4: \( \sqrt{x} \)**

- Rewrite: \( x^{1/2} \)
- The term \( x^{1/2} \) has a fractional exponent, which violates the definition of a polynomial function.

**Conclusion:** This is NOT a polynomial function.

### Final Answer
The only option that qualifies as a polynomial function is **Option 3: \( 3x^2 + 2x + 1 \)**.

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