https://r2.erweima.ai/mathbot/image/1727223156332j0nvk4tl.jpg

resuelve GA= grado absoluto GR= grado relativo 

Let's go through the problems one by one based on the image you've provided:

### 6. Problem:  
We are given that \( P(x, y) = 7x^{a+2}y^{b+3} + \sqrt{5}x^{a+3}y^{b+1} \) and the relative degrees are \( GR(x) = 7 \) and \( GR(y) = 5 \). We are asked to find the absolute degree (GA).

- **Relative degree in \(x\), GR(x):** The relative degree in \(x\) is the highest power of \(x\) in the polynomial, so:
  - From the first term: \( 7x^{a+2}y^{b+3} \), the power of \(x\) is \(a+2\).
  - From the second term: \( \sqrt{5}x^{a+3}y^{b+1} \), the power of \(x\) is \(a+3\).

  The highest degree in \(x\) is \(a+3\), and it’s given that \( GR(x) = 7 \), so:
  \[
  a+3 = 7 \quad \Rightarrow \quad a = 4.
  \]

- **Relative degree in \(y\), GR(y):** The relative degree in \(y\) is the highest power of \(y\):
  - From the first term: \( 7x^{a+2}y^{b+3} \), the power of \(y\) is \(b+3\).
  - From the second term: \( \sqrt{5}x^{a+3}y^{b+1} \), the power of \(y\) is \(b+1\).

  The highest degree in \(y\) is \(b+3\), and it’s given that \( GR(y) = 5 \), so:
  \[
  b+3 = 5 \quad \Rightarrow \quad b = 2.
  \]

- **Absolute degree (GA):** The absolute degree is the sum of the degrees of both variables in the term with the highest total degree. 
  - For the first term \( 7x^{a+2}y^{b+3} = 7x^6y^5 \), the total degree is \(6 + 5 = 11\).
  - For the second term \( \sqrt{5}x^{a+3}y^{b+1} = \sqrt{5}x^7y^3 \), the total degree is \(7 + 3 = 10\).

  The highest total degree is 11, so the absolute degree is:
  \[
  GA = 11.
  \]

### 7. Problem:
We are given \( Q(x) = 4x^{b+3} - 5x^{b+1} + 7x^{b+5} \) and \( GA = 13 \), and we need to find \(b\).

- The absolute degree is determined by the highest power of \(x\), which is \(b+5\) from the third term.
  
  We know the absolute degree is 13, so:
  \[
  b + 5 = 13 \quad \Rightarrow \quad b = 8.
  \]

### 8. Problem:
We are given \( P(x, y) = x^{m+3} - 4x^my^n + 2x^{m+2}y^{n-1} \), with relative degrees \( GR(x) = 5 \) and \( GR(y) = 4 \). We need to calculate \( m - n \).

- **Relative degree in \(x\), GR(x):** The highest degree in \(x\) is \(m+3\) from the first term. We know \( GR(x) = 5 \), so:
  \[
  m+3 = 5 \quad \Rightarrow \quad m = 2.
  \]

- **Relative degree in \(y\), GR(y):** The highest degree in \(y\) is \(n\) from the second term. We know \( GR(y) = 4 \), so:
  \[
  n = 4.
  \]

Now, we calculate \(m - n\):
\[
m - n = 2 - 4 = -2.
\]

### Summary of Answers:
1. Problem 6: \( GA = 11 \)
2. Problem 7: \( b = 8 \)
3. Problem 8: \( m - n = -2 \)

Would you like further explanations or have any questions? Here are 5 related questions to consider:

1. How do you find the absolute degree of a polynomial in multiple variables?
2. What is the difference between relative and absolute degrees in polynomials?
3. How does changing the coefficients of terms affect the degree of a polynomial?
4. Can you provide another example where both the relative degree in \(x\) and \(y\) are given?
5. How would you approach finding the degree of a polynomial with more than two variables?

**Tip:** Always check both the highest degree in \(x\) and \(y\) when finding relative and absolute degrees.

Resuelva los problemas de grados relativos y grados absolutos basados en las polinomios dados.

Learn how to calculate the relative and absolute degrees of polynomial equations in two variables. This step-by-step solution helps you solve for unknowns and determine the highest degree terms in the polynomial. Suitable for Grades 10-12 students.

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