Identify which of the following algebraic expressions are polynomials. If so, write their degrees.
`(2)/(3)x^(2)-(5)/(7)x-(1)/(4)`

Answer

To determine if 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:
\[ \frac{2}{3}x^2 - \frac{5}{7}x - \frac{1}{4} \]

### Step 2: Check the powers of the variable
In this expression, we need to check the powers of the variable \(x\):
- The first term is \(\frac{2}{3}x^2\), which has a power of \(2\).
- The second term is \(-\frac{5}{7}x\), which has a power of \(1\) (since \(x\) is the same as \(x^1\)).
- The third term is \(-\frac{1}{4}\), which is a constant term and can be considered as \(x^0\) (since any number to the power of 0 is 1).

### Step 3: Determine if it is a polynomial
For an expression to be a polynomial:
- The powers of the variable must be non-negative integers (0, 1, 2, ...).
- All coefficients can be any real numbers.

Since all the powers of \(x\) in the expression are non-negative integers (2, 1, and 0), the expression is indeed a polynomial.

### Step 4: Find the degree of the polynomial
The degree of a polynomial is defined as the highest power of the variable in the expression. In this case:
- The highest power of \(x\) is \(2\) (from the term \(\frac{2}{3}x^2\)).

Thus, the degree of the polynomial is \(2\).

### Final Conclusion
The expression \(\frac{2}{3}x^2 - \frac{5}{7}x - \frac{1}{4}\) is a polynomial, and its degree is \(2\).

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