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Sagot :
To determine the degrees of the sum and the difference of the given polynomials, let's first simplify the sum and the difference step-by-step.
### Given Polynomials:
[tex]\[ P(x, y) = 3x^5 y - 2x^3 y^4 - 7x y^3 \][/tex]
[tex]\[ Q(x, y) = -8x^5 y + 2x^3 y^4 + x y^3 \][/tex]
### Step-by-Step Solution:
#### Sum of the Polynomials:
First, add the polynomials [tex]\( P(x, y) + Q(x, y) \)[/tex]:
[tex]\[ P(x, y) + Q(x, y) = (3x^5 y - 2x^3 y^4 - 7x y^3) + (-8x^5 y + 2x^3 y^4 + x y^3) \][/tex]
Combine like terms:
[tex]\[ (3x^5 y - 8x^5 y) + (-2x^3 y^4 + 2x^3 y^4) + (-7x y^3 + x y^3) \][/tex]
Simplify:
[tex]\[ -5x^5 y + 0x^3 y^4 - 6x y^3 \][/tex]
[tex]\[ -5x^5 y - 6x y^3 \][/tex]
Now, determine the degree of the resulting polynomial. The degree of a polynomial is the highest power of the terms. In this sum, we have two terms:
[tex]\[ -5x^5 y \quad \text{(degree: } 5 + 1 = 6\text{)} \][/tex]
[tex]\[ -6x y^3 \quad \text{(degree: } 1 + 3 = 4\text{)} \][/tex]
The highest degree term is [tex]\(-5x^5 y\)[/tex] which has a degree of 6.
So, the degree of the sum is 6.
#### Difference of the Polynomials:
Next, subtract the polynomials [tex]\( P(x, y) - Q(x, y) \)[/tex]:
[tex]\[ P(x, y) - Q(x, y) = (3x^5 y - 2x^3 y^4 - 7x y^3) - (-8x^5 y + 2x^3 y^4 + x y^3) \][/tex]
Distribute the subtraction:
[tex]\[ (3x^5 y - 2x^3 y^4 - 7x y^3) + (8x^5 y - 2x^3 y^4 - x y^3) \][/tex]
Combine like terms:
[tex]\[ (3x^5 y + 8x^5 y) + (-2x^3 y^4 - 2x^3 y^4) + (-7x y^3 - x y^3) \][/tex]
Simplify:
[tex]\[ 11x^5 y - 4x^3 y^4 - 8x y^3 \][/tex]
Now determine the degree of the resulting polynomial. In this difference, we have three terms:
[tex]\[ 11x^5 y \quad \text{(degree: } 5 + 1 = 6\text{)} \][/tex]
[tex]\[ -4x^3 y^4 \quad \text{(degree: } 3 + 4 = 7\text{)} \][/tex]
[tex]\[ -8x y^3 \quad \text{(degree: } 1 + 3 = 4\text{)} \][/tex]
The highest degree term is [tex]\(-4x^3 y^4\)[/tex] which has a degree of 7.
So, the degree of the difference is 7.
### Conclusion:
The sum of the given polynomials has a degree of 6, and the difference of the given polynomials has a degree of 7.
Therefore, the correct statement is:
The sum has a degree of 6, but the difference has a degree of 7.
### Given Polynomials:
[tex]\[ P(x, y) = 3x^5 y - 2x^3 y^4 - 7x y^3 \][/tex]
[tex]\[ Q(x, y) = -8x^5 y + 2x^3 y^4 + x y^3 \][/tex]
### Step-by-Step Solution:
#### Sum of the Polynomials:
First, add the polynomials [tex]\( P(x, y) + Q(x, y) \)[/tex]:
[tex]\[ P(x, y) + Q(x, y) = (3x^5 y - 2x^3 y^4 - 7x y^3) + (-8x^5 y + 2x^3 y^4 + x y^3) \][/tex]
Combine like terms:
[tex]\[ (3x^5 y - 8x^5 y) + (-2x^3 y^4 + 2x^3 y^4) + (-7x y^3 + x y^3) \][/tex]
Simplify:
[tex]\[ -5x^5 y + 0x^3 y^4 - 6x y^3 \][/tex]
[tex]\[ -5x^5 y - 6x y^3 \][/tex]
Now, determine the degree of the resulting polynomial. The degree of a polynomial is the highest power of the terms. In this sum, we have two terms:
[tex]\[ -5x^5 y \quad \text{(degree: } 5 + 1 = 6\text{)} \][/tex]
[tex]\[ -6x y^3 \quad \text{(degree: } 1 + 3 = 4\text{)} \][/tex]
The highest degree term is [tex]\(-5x^5 y\)[/tex] which has a degree of 6.
So, the degree of the sum is 6.
#### Difference of the Polynomials:
Next, subtract the polynomials [tex]\( P(x, y) - Q(x, y) \)[/tex]:
[tex]\[ P(x, y) - Q(x, y) = (3x^5 y - 2x^3 y^4 - 7x y^3) - (-8x^5 y + 2x^3 y^4 + x y^3) \][/tex]
Distribute the subtraction:
[tex]\[ (3x^5 y - 2x^3 y^4 - 7x y^3) + (8x^5 y - 2x^3 y^4 - x y^3) \][/tex]
Combine like terms:
[tex]\[ (3x^5 y + 8x^5 y) + (-2x^3 y^4 - 2x^3 y^4) + (-7x y^3 - x y^3) \][/tex]
Simplify:
[tex]\[ 11x^5 y - 4x^3 y^4 - 8x y^3 \][/tex]
Now determine the degree of the resulting polynomial. In this difference, we have three terms:
[tex]\[ 11x^5 y \quad \text{(degree: } 5 + 1 = 6\text{)} \][/tex]
[tex]\[ -4x^3 y^4 \quad \text{(degree: } 3 + 4 = 7\text{)} \][/tex]
[tex]\[ -8x y^3 \quad \text{(degree: } 1 + 3 = 4\text{)} \][/tex]
The highest degree term is [tex]\(-4x^3 y^4\)[/tex] which has a degree of 7.
So, the degree of the difference is 7.
### Conclusion:
The sum of the given polynomials has a degree of 6, and the difference of the given polynomials has a degree of 7.
Therefore, the correct statement is:
The sum has a degree of 6, but the difference has a degree of 7.
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