Idenitfy which of the following algebraic expression are polynomials. If so, write their degrees.
`(2)/(5)x^(3)-7x^(2)-(1)/(2)x+3`

Answer

To determine whether the given algebraic expression is a polynomial and to find its degree, we can follow these steps:

### Step 1: Identify the Expression
The given expression is:
\[ \frac{2}{5}x^3 - 7x^2 - \frac{1}{2}x + 3 \]

### Step 2: Check for Polynomial Criteria
A polynomial is an expression that consists of variables raised to non-negative integer powers, combined with coefficients (which can be constants). The expression must not have any variables in the denominator, nor any negative or fractional exponents.

- The terms in the expression are:
- \(\frac{2}{5}x^3\) (where \(x\) is raised to the power of 3)
- \(-7x^2\) (where \(x\) is raised to the power of 2)
- \(-\frac{1}{2}x\) (where \(x\) is raised to the power of 1)
- \(3\) (a constant term)

All the exponents (3, 2, 1, and 0 for the constant) are non-negative integers, and there are no variables in the denominator.

### Step 3: Conclusion about Polynomial Status
Since all terms meet the criteria for being a polynomial, we conclude that:
\[ \frac{2}{5}x^3 - 7x^2 - \frac{1}{2}x + 3 \text{ is a polynomial.} \]

### Step 4: Determine the Degree of the Polynomial
The degree of a polynomial is the highest exponent of the variable in the expression.

- The exponents in the expression are:
- \(3\) from \(\frac{2}{5}x^3\)
- \(2\) from \(-7x^2\)
- \(1\) from \(-\frac{1}{2}x\)
- \(0\) from \(3\) (constant term)

The highest exponent is \(3\).

### Final Answer
Thus, the degree of the polynomial is:
\[ \text{Degree} = 3 \]

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