Identify which of the following expressions are polynomials. If so, write their degrees.
`ab^(2)-(7)/(a^(2))+5b^(2)+6`

Answer

To determine whether the expression \( ab^2 - \frac{7}{a^2} + 5b^2 + 6 \) is a polynomial and to find its degree, we will analyze each term in the expression.

### Step 1: Identify the terms in the expression
The expression is:
\[ ab^2 - \frac{7}{a^2} + 5b^2 + 6 \]

The terms are:
1. \( ab^2 \)
2. \( -\frac{7}{a^2} \)
3. \( 5b^2 \)
4. \( 6 \)

### Step 2: Check each term for polynomial criteria
A polynomial is an expression that consists of variables raised to non-negative integer powers.

- **Term 1: \( ab^2 \)**
- Here, \( a \) is raised to the power of 1 (which is non-negative) and \( b \) is raised to the power of 2 (which is also non-negative). This term is a polynomial term.

- **Term 2: \( -\frac{7}{a^2} \)**
- This term can be rewritten as \( -7a^{-2} \). The variable \( a \) is raised to a negative power (-2), which does not satisfy the criteria for a polynomial. Therefore, this term is not a polynomial term.

- **Term 3: \( 5b^2 \)**
- Here, \( b \) is raised to the power of 2 (which is non-negative). This term is a polynomial term.

- **Term 4: \( 6 \)**
- This is a constant term and can be considered as \( 6b^0 \) (where \( b \) is raised to the power of 0). This term is also a polynomial term.

### Step 3: Conclusion
Since the expression contains a term \( -\frac{7}{a^2} \) that is not a polynomial, the entire expression is not a polynomial.

### Final Answer
The expression \( ab^2 - \frac{7}{a^2} + 5b^2 + 6 \) is **not a polynomial** because it contains a term with a negative exponent.

---