Identify which of the following expressions are polynomials. If so, write their degrees.
`(2)/(3)x^(4)-7x+15`

Answer

To determine if the expression \( \frac{2}{3}x^4 - 7x + 15 \) is a polynomial and to find its degree, we can follow these steps:

### Step 1: Identify the expression
The given expression is:
\[ \frac{2}{3}x^4 - 7x + 15 \]

### Step 2: Check if it is a polynomial
A polynomial is an expression that consists of variables raised to non-negative integer powers and coefficients that are real numbers.

- The term \( \frac{2}{3}x^4 \) has a variable \( x \) raised to the power of 4, which is a non-negative integer.
- The term \( -7x \) has a variable \( x \) raised to the power of 1, which is also a non-negative integer.
- The term \( 15 \) is a constant, which can be considered as \( 15x^0 \) (where \( x^0 = 1 \)).

Since all terms in the expression meet the criteria for being part of a polynomial, we can conclude that:
\[ \frac{2}{3}x^4 - 7x + 15 \text{ is a polynomial.} \]

### Step 3: Determine the degree of the polynomial
The degree of a polynomial is defined as the highest power of the variable in the expression.

- The highest power of \( x \) in the expression \( \frac{2}{3}x^4 - 7x + 15 \) is \( x^4 \).

Thus, the degree of the polynomial is:
\[ \text{Degree} = 4 \]

### Final Conclusion
The expression \( \frac{2}{3}x^4 - 7x + 15 \) is a polynomial, and its degree is 4.

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