Idenitfy which of the following algebraic expression are polynomials. If so, write their degrees.
`3x^(2)y-(2)/(xy)+5xy^(2)`

Answer

To determine whether the given expression \(3x^2y - \frac{2}{xy} + 5xy^2\) is a polynomial, we need to analyze each term in the expression and check if they meet the criteria for being a polynomial.

### Step-by-Step Solution:

1. **Identify the Terms in the Expression**:
The expression consists of three terms:
- \(3x^2y\)
- \(-\frac{2}{xy}\)
- \(5xy^2\)

2. **Check Each Term**:
- **First Term: \(3x^2y\)**:
- This term can be expressed as \(3x^2y^1\).
- The exponents of \(x\) and \(y\) are both non-negative integers (2 and 1 respectively).
- Therefore, this term is a polynomial term.

- **Second Term: \(-\frac{2}{xy}\)**:
- This term can be rewritten as \(-2x^{-1}y^{-1}\).
- The exponents of \(x\) and \(y\) are both negative (-1 for both).
- Since polynomials cannot have negative exponents, this term is **not** a polynomial term.

- **Third Term: \(5xy^2\)**:
- This term can be expressed as \(5x^1y^2\).
- The exponents of \(x\) and \(y\) are both non-negative integers (1 and 2 respectively).
- Therefore, this term is a polynomial term.

3. **Conclusion**:
Since one of the terms (\(-\frac{2}{xy}\)) is not a polynomial term due to its negative exponents, the entire expression \(3x^2y - \frac{2}{xy} + 5xy^2\) is **not** a polynomial.

### Summary of Degrees:
- The degrees of the polynomial terms are:
- For \(3x^2y\): Degree = \(2 + 1 = 3\)
- For \(5xy^2\): Degree = \(1 + 2 = 3\)

However, since the expression contains a non-polynomial term, we conclude that:

**The expression \(3x^2y - \frac{2}{xy} + 5xy^2\) is not a polynomial.**