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The exercise asks you to determine the degree, the degree with respect to \(x\), and the degree with respect to \(y\) of the following polynomials. Let's analyze each polynomial step by step.

### Polynomial (a) \( 3x^4y^2 + 2xy^5 \)
- **Degree with respect to \(x\)**: The degree is the highest power of \(x\) in any term. For \(3x^4y^2\), the degree of \(x\) is 4. For \(2xy^5\), the degree of \(x\) is 1. Therefore, the degree with respect to \(x\) is **4**.
- **Degree with respect to \(y\)**: The degree is the highest power of \(y\) in any term. For \(3x^4y^2\), the degree of \(y\) is 2. For \(2xy^5\), the degree of \(y\) is 5. Therefore, the degree with respect to \(y\) is **5**.
- **Total Degree**: The total degree of the polynomial is the highest sum of the exponents of \(x\) and \(y\) in any term. For \(3x^4y^2\), the sum is \(4 + 2 = 6\). For \(2xy^5\), the sum is \(1 + 5 = 6\). Therefore, the total degree is **6**.

### Polynomial (b) \( 3x^2 - 2x^2y^3 + 4xy^5 \)
- **Degree with respect to \(x\)**: For \(3x^2\), the degree of \(x\) is 2. For \(-2x^2y^3\), it is 2. For \(4xy^5\), it is 1. Therefore, the degree with respect to \(x\) is **2**.
- **Degree with respect to \(y\)**: For \(3x^2\), the degree of \(y\) is 0. For \(-2x^2y^3\), it is 3. For \(4xy^5\), it is 5. Therefore, the degree with respect to \(y\) is **5**.
- **Total Degree**: For \(3x^2\), the sum is \(2 + 0 = 2\). For \(-2x^2y^3\), it is \(2 + 3 = 5\). For \(4xy^5\), it is \(1 + 5 = 6\). Therefore, the total degree is **6**.

### Polynomial (c) \( x^7 + xy^2 \)
- **Degree with respect to \(x\)**: For \(x^7\), the degree of \(x\) is 7. For \(xy^2\), it is 1. Therefore, the degree with respect to \(x\) is **7**.
- **Degree with respect to \(y\)**: For \(x^7\), the degree of \(y\) is 0. For \(xy^2\), it is 2. Therefore, the degree with respect to \(y\) is **2**.
- **Total Degree**: For \(x^7\), the sum is \(7 + 0 = 7\). For \(xy^2\), it is \(1 + 2 = 3\). Therefore, the total degree is **7**.

### Polynomial (d) \( x^3y^2 - 2x^2y^4 \)
- **Degree with respect to \(x\)**: For \(x^3y^2\), the degree of \(x\) is 3. For \(-2x^2y^4\), it is 2. Therefore, the degree with respect to \(x\) is **3**.
- **Degree with respect to \(y\)**: For \(x^3y^2\), the degree of \(y\) is 2. For \(-2x^2y^4\), it is 4. Therefore, the degree with respect to \(y\) is **4**.
- **Total Degree**: For \(x^3y^2\), the sum is \(3 + 2 = 5\). For \(-2x^2y^4\), it is \(2 + 4 = 6\). Therefore, the total degree is **6**.

### Polynomial (e) \( 2xy + 3x^2y - y^7 \)
- **Degree with respect to \(x\)**: For \(2xy\), the degree of \(x\) is 1. For \(3x^2y\), it is 2. For \(-y^7\), it is 0. Therefore, the degree with respect to \(x\) is **2**.
- **Degree with respect to \(y\)**: For \(2xy\), the degree of \(y\) is 1. For \(3x^2y\), it is 1. For \(-y^7\), it is 7. Therefore, the degree with respect to \(y\) is **7**.
- **Total Degree**: For \(2xy\), the sum is \(1 + 1 = 2\). For \(3x^2y\), it is \(2 + 1 = 3\). For \(-y^7\), it is \(0 + 7 = 7\). Therefore, the total degree is **7**.

### Polynomial (f) \( -7x^5y^4 + 2x^2y^3 - xy \)
- **Degree with respect to \(x\)**: For \(-7x^5y^4\), the degree of \(x\) is 5. For \(2x^2y^3\), it is 2. For \(-xy\), it is 1. Therefore, the degree with respect to \(x\) is **5**.
- **Degree with respect to \(y\)**: For \(-7x^5y^4\), the degree of \(y\) is 4. For \(2x^2y^3\), it is 3. For \(-xy\), it is 1. Therefore, the degree with respect to \(y\) is **4**.
- **Total Degree**: For \(-7x^5y^4\), the sum is \(5 + 4 = 9\). For \(2x^2y^3\), it is \(2 + 3 = 5\). For \(-xy\), it is \(1 + 1 = 2\). Therefore, the total degree is **9**.

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Would you like more details on any step or have any other questions?

Here are 5 related questions you might consider:
1. How does the degree of a polynomial affect its graph in two variables?
2. What is the significance of the total degree of a polynomial in multiple variables?
3. How would the degree of the polynomial change if you added or subtracted two polynomials?
4. Can you explain how the degree affects the behavior of a polynomial function at infinity?
5. How would you calculate the degree of a polynomial in more than two variables?

**Tip:** Remember, the degree of a polynomial is a critical factor that influences its properties, such as the number of roots and the general shape of its graph.

Explore how to determine the degree of polynomials in two variables (x and y) through detailed examples. Learn about the significance of degree in polynomial functions and how it affects their properties. Suitable for Grades 9-12 students studying algebra and multivariable functions.

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