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

Answer

To determine if the given algebraic expression \( x^2y - 2xy + \frac{2}{3}x - 1 \) is a polynomial and to find its degree, we will follow these steps:

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

### Step 2: Check if it is a polynomial
A polynomial is defined as an algebraic expression where all the powers of the variables are whole numbers (non-negative integers).

- In the first term \( x^2y \):
- The power of \( x \) is \( 2 \) (which is a whole number).
- The power of \( y \) is \( 1 \) (which is also a whole number).

- In the second term \( -2xy \):
- The power of \( x \) is \( 1 \).
- The power of \( y \) is \( 1 \).

- In the third term \( \frac{2}{3}x \):
- The power of \( x \) is \( 1 \) (and there is no \( y \) term).

- In the fourth term \( -1 \):
- This is a constant term, which can be considered as \( x^0y^0 \).

Since all the powers of the variables in the expression are whole numbers, we conclude that this expression is indeed a polynomial.

### Step 3: Find the degree of the polynomial
The degree of a polynomial is the highest sum of the powers of the variables in any single term.

- For the term \( x^2y \):
- The sum of the powers is \( 2 + 1 = 3 \).

- For the term \( -2xy \):
- The sum of the powers is \( 1 + 1 = 2 \).

- For the term \( \frac{2}{3}x \):
- The sum of the powers is \( 1 \).

- For the constant term \( -1 \):
- The sum of the powers is \( 0 \).

Now, we compare the sums:
- The highest sum of the powers is \( 3 \) from the term \( x^2y \).

Thus, the degree of the polynomial is \( 3 \).

### Final Answer
The expression \( x^2y - 2xy + \frac{2}{3}x - 1 \) is a polynomial, and its degree is \( 3 \).

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