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Let's analyze the polynomial \( P(x, y) = 6x^8y - 3x^7y^3 + 2xy^5 \).

### 1. Grado relativo respecto a \( x \) (Relative degree with respect to \( x \)):
- **First term:** \( 6x^8y \), the degree in \( x \) is 8.
- **Second term:** \( -3x^7y^3 \), the degree in \( x \) is 7.
- **Third term:** \( 2xy^5 \), the degree in \( x \) is 1.

The **relative degree with respect to \( x \)** is the highest degree of \( x \) in the polynomial, which is **8**.

### 2. Grado relativo respecto a \( y \) (Relative degree with respect to \( y \)):
- **First term:** \( 6x^8y \), the degree in \( y \) is 1.
- **Second term:** \( -3x^7y^3 \), the degree in \( y \) is 3.
- **Third term:** \( 2xy^5 \), the degree in \( y \) is 5.

The **relative degree with respect to \( y \)** is the highest degree of \( y \) in the polynomial, which is **5**.

### 3. Grado absoluto (Absolute degree):
The absolute degree is the highest sum of the exponents of \( x \) and \( y \) in any term.
- **First term:** \( 6x^8y \), the sum of the exponents is \( 8 + 1 = 9 \).
- **Second term:** \( -3x^7y^3 \), the sum of the exponents is \( 7 + 3 = 10 \).
- **Third term:** \( 2xy^5 \), the sum of the exponents is \( 1 + 5 = 6 \).

The **absolute degree** of the polynomial is the highest of these sums, which is **10**.

### 4. Coeficiente principal (Leading coefficient):
The leading coefficient is the coefficient of the term with the highest absolute degree.
- In this case, the term with the highest absolute degree is \( -3x^7y^3 \), and its coefficient is **-3**.

### Summary:
- **Relative degree with respect to \( x \):** 8
- **Relative degree with respect to \( y \):** 5
- **Absolute degree:** 10
- **Leading coefficient:** -3

Would you like further details or have any questions about this solution?

Here are 5 related questions you might find interesting:
1. How would the degrees change if an additional term like \( 4x^2y^4 \) were added to the polynomial?
2. What is the significance of the leading coefficient in the behavior of the polynomial?
3. How do you determine the degree of a polynomial with more than two variables?
4. What happens to the absolute degree if all the terms have the same sum of exponents?
5. Can a polynomial have the same absolute degree but different leading coefficients?

**Tip:** When working with polynomials, always identify the term with the highest total degree to easily determine the absolute degree and leading coefficient.

Halle el grado relativo de x; y; el grado absoluto y el coeficiente principal de: P(x, y) = 6x^8y - 3x^7y^3 + 2xy^5

Learn how to find the relative degrees with respect to x and y, the absolute degree, and the leading coefficient of the polynomial P(x, y) = 6x^8y - 3x^7y^3 + 2xy^5. This problem covers key algebraic concepts suitable for students in Grades 10-12.

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