Identify which of the following expressions are polynomials. If so, write their degrees.
`4x^(5)-7x^(5)y+3xy^(4) +8y^(5)`

Answer

To determine whether the given expression \(4x^{5} - 7x^{5}y + 3xy^{4} + 8y^{5}\) is a polynomial and to find its degree, we can follow these steps:

### Step 1: Identify the terms in the expression
The expression consists of the following terms:
1. \(4x^{5}\)
2. \(-7x^{5}y\)
3. \(3xy^{4}\)
4. \(8y^{5}\)

### Step 2: Check if each term is a polynomial term
A polynomial term is defined as a term that has non-negative integer exponents for its variables. Let's check each term:

1. **Term \(4x^{5}\)**: The exponent of \(x\) is \(5\) (which is non-negative). This is a polynomial term.
2. **Term \(-7x^{5}y\)**: The exponent of \(x\) is \(5\) and the exponent of \(y\) is \(1\) (both are non-negative). This is a polynomial term.
3. **Term \(3xy^{4}\)**: The exponent of \(x\) is \(1\) and the exponent of \(y\) is \(4\) (both are non-negative). This is a polynomial term.
4. **Term \(8y^{5}\)**: The exponent of \(y\) is \(5\) (which is non-negative). This is a polynomial term.

### Step 3: Conclusion about the expression
Since all the terms in the expression have non-negative integer exponents, we can conclude that the entire expression is a polynomial.

### Step 4: Find the degree of the polynomial
The degree of a polynomial is the highest exponent of its variables when the polynomial is expressed in standard form. We need to find the degree of each term:

1. **Degree of \(4x^{5}\)**: \(5\)
2. **Degree of \(-7x^{5}y\)**: \(5 + 1 = 6\) (sum of the exponents)
3. **Degree of \(3xy^{4}\)**: \(1 + 4 = 5\) (sum of the exponents)
4. **Degree of \(8y^{5}\)**: \(5\)

The highest degree among these terms is \(6\).

### Final Answer
The expression \(4x^{5} - 7x^{5}y + 3xy^{4} + 8y^{5}\) is a polynomial, and its degree is \(6\).

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