Identify which of the following algebraic expressions are polynomials. If so, write their degrees.
`3x^(2)+(2)/(x)-3`

Answer

To determine whether the given algebraic expression \(3x^2 + \frac{2}{x} - 3\) is a polynomial and to find its degree if it is, we will follow these steps:

### Step 1: Identify the expression
The expression we are analyzing is:
\[ 3x^2 + \frac{2}{x} - 3 \]

### Step 2: Rewrite the expression
We can rewrite the term \(\frac{2}{x}\) as \(2x^{-1}\). Therefore, the expression becomes:
\[ 3x^2 + 2x^{-1} - 3 \]

### Step 3: Analyze the powers of the variables
Now, we will look at the powers of \(x\) in each term:
- The first term \(3x^2\) has a power of \(2\).
- The second term \(2x^{-1}\) has a power of \(-1\).
- The third term \(-3\) can be considered as \(-3x^0\), which has a power of \(0\).

### Step 4: Check for polynomial conditions
For an expression to be a polynomial, all the powers of the variable must be whole numbers (non-negative integers). In our case:
- The power \(2\) (from \(3x^2\)) is a whole number.
- The power \(-1\) (from \(2x^{-1}\)) is not a whole number.
- The power \(0\) (from \(-3\)) is a whole number.

### Step 5: Conclusion
Since one of the terms (\(2x^{-1}\)) has a negative power, the expression \(3x^2 + \frac{2}{x} - 3\) is **not a polynomial**.

### Final Answer
The expression \(3x^2 + \frac{2}{x} - 3\) is not a polynomial.

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