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Which is true about the degree of the sum and difference of the polynomials [tex]\(3x^5y - 2x^3y^4 - 7xy^3\)[/tex] and [tex]\(-8x^5y + 2x^3y^4 + xy^3\)[/tex]?

A. Both the sum and difference have a degree of 6.
B. Both the sum and difference have a degree of 7.
C. The sum has a degree of 6, but the difference has a degree of 7.
D. The sum has a degree of 7, but the difference has a degree of 6.

Sagot :

GinnyAnswer
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.
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