Let's solve the problem step-by-step:
### Step 1: Define the Polynomials
The two given polynomials are:
[tex]\[ P(x, y) = 3x^2 y^2 - 2xy^5 \][/tex]
[tex]\[ Q(x, y) = -3x^2 y^2 + 3x^4 y \][/tex]
### Step 2: Sum the Polynomials
We need to add these two polynomials:
[tex]\[
P(x, y) + Q(x, y) = (3x^2 y^2 - 2xy^5) + (-3x^2 y^2 + 3x^4 y)
\][/tex]
### Step 3: Combine Like Terms
Combine like terms in the expression:
[tex]\[
(3x^2 y^2 - 3x^2 y^2) + (-2xy^5 + 3x^4 y)
\][/tex]
Simplifying each group of like terms:
[tex]\[
3x^2 y^2 - 3x^2 y^2 = 0
\][/tex]
[tex]\[
-2xy^5 + 3x^4 y = -2xy^5 + 3x^4 y
\][/tex]
So, the simplified sum of the polynomials is:
[tex]\[
-2xy^5 + 3x^4 y
\][/tex]
### Step 4: Classify the Resultant Polynomial
The simplified polynomial [tex]\(-2xy^5 + 3x^4 y\)[/tex] has two distinct terms, so it is a binomial.
### Step 5: Determine the Degree
The degree of a polynomial term is the sum of the exponents of its variables. There are two terms to consider:
1. [tex]\(-2xy^5\)[/tex]:
[tex]\[
\text{Degree} = 1 (for x) + 5 (for y) = 6
\][/tex]
2. [tex]\(3x^4 y\)[/tex]:
[tex]\[
\text{Degree} = 4 (for x) + 1 (for y) = 5
\][/tex]
The highest degree among the terms is 6.
### Conclusion
The completely simplified sum of the polynomials [tex]\(3x^2 y^2 - 2xy^5\)[/tex] and [tex]\(-3x^2 y^2 + 3x^4 y\)[/tex] is a binomial with a degree of 6.
Thus, the correct answer is:
[tex]\[
\text{The sum is a binomial with a degree of 6.}
\][/tex]
