To determine the true nature of the completely simplified sum of the polynomials [tex]\(3 x^2 y^2 - 2 x y^5\)[/tex] and [tex]\(-3 x^2 y^2 + 3 x^4 y\)[/tex], let's analyze and simplify the given polynomials step-by-step. We will then determine the number of terms (whether it is a binomial or trinomial) and the degree of the simplified expression.
### Step 1: Combine the Polynomials
Given polynomials:
[tex]\[ P_1 = 3 x^2 y^2 - 2 x y^5 \][/tex]
[tex]\[ P_2 = -3 x^2 y^2 + 3 x^4 y \][/tex]
Now, add the polynomials together:
[tex]\[ P_{\text{sum}} = P_1 + P_2 \][/tex]
[tex]\[ P_{\text{sum}} = (3 x^2 y^2 - 2 x y^5) + (-3 x^2 y^2 + 3 x^4 y) \][/tex]
### Step 2: Simplify the Addition
Combine like terms to simplify:
[tex]\[ P_{\text{sum}} = (3 x^2 y^2 - 3 x^2 y^2) + (3 x^4 y - 2 x y^5) \][/tex]
[tex]\[ P_{\text{sum}} = 0 + (3 x^4 y - 2 x y^5) \][/tex]
[tex]\[ P_{\text{sum}} = 3 x^4 y - 2 x y^5 \][/tex]
### Step 3: Factor and Simplify Further (if Possible)
Factor out the common term [tex]\(x y\)[/tex]:
[tex]\[ P_{\text{sum}} = x y (3 x^3 - 2 y^4) \][/tex]
### Step 4: Determine the Number of Terms
The expression [tex]\( x y (3 x^3 - 2 y^4) \)[/tex] is a product of [tex]\(x y\)[/tex] and a binomial [tex]\( (3 x^3 - 2 y^4) \)[/tex]. Hence, [tex]\(x y (3 x^3 - 2 y^4)\)[/tex] is a binomial, where there are two distinct monomials being multiplied by [tex]\( x y \)[/tex].
### Step 5: Determine the Degree of the Polynomial
The degree of a polynomial is the highest sum of the exponents of the variables in any single term. We need to check the terms in:
[tex]\[ 3 x^4 y \quad \text{and} \quad -2 x y^5 \][/tex]
For [tex]\(3 x^4 y\)[/tex]:
- The degree is [tex]\(4 + 1 = 5\)[/tex].
For [tex]\(-2 x y^5\)[/tex]:
- The degree is [tex]\(1 + 5 = 6\)[/tex].
The highest degree among the terms is [tex]\(6\)[/tex].
### Conclusion
The completely simplified polynomial is:
[tex]\[ x y (3 x^3 - 2 y^4) \][/tex]
- It is a binomial.
- The degree of the polynomial is [tex]\(6\)[/tex].
So, the correct statement about the polynomial is:
- The sum is a binomial with a degree of 6.
