Step-by-Step Solution:
To find the completely simplified sum of the given polynomials, let's start by writing the two polynomials clearly and then sum them.
Given polynomials:
[tex]\[ p_1 = 3x^2 y^2 - 2xy^5 \][/tex]
[tex]\[ p_2 = -3x^2 y^2 + 3x^4 y \][/tex]
Step 1: Sum the polynomials
We sum [tex]\( p_1 \)[/tex] and [tex]\( p_2 \)[/tex]:
[tex]\[ (3x^2 y^2 - 2xy^5) + (-3x^2 y^2 + 3x^4 y) \][/tex]
Combine like terms:
[tex]\[ (3x^2 y^2 - 3x^2 y^2) + 3x^4 y - 2xy^5 \][/tex]
Notice that [tex]\( 3x^2 y^2 \)[/tex] and [tex]\( -3x^2 y^2 \)[/tex] cancel each other out:
[tex]\[ 0 + 3x^4 y - 2xy^5 \][/tex]
So, the resulting polynomial is:
[tex]\[ 3x^4 y - 2xy^5 \][/tex]
Step 2: Analyze the resulting polynomial
We need to determine the number of terms, degree of the polynomial, and its type.
Step 3: Count the number of terms
The resulting polynomial [tex]\( 3x^4 y - 2xy^5 \)[/tex] has two terms: [tex]\( 3x^4 y \)[/tex] and [tex]\( -2xy^5 \)[/tex].
Step 4: Determine the degree of the polynomial
The degree of a polynomial is the highest degree of its terms.
For the term [tex]\( 3x^4 y \)[/tex]:
- The degree is [tex]\( 4 + 1 = 5 \)[/tex].
For the term [tex]\( -2xy^5 \)[/tex]:
- The degree is [tex]\( 1 + 5 = 6 \)[/tex].
So, the highest degree among the terms [tex]\( 3x^4 y \)[/tex] and [tex]\( -2xy^5 \)[/tex] is 6.
Step 5: Conclusion
The resulting polynomial [tex]\( 3x^4 y - 2xy^5 \)[/tex] is a binomial (since it has two terms) with a degree of 6.
Thus, the statement that is true about the completely simplified sum of the polynomials is:
[tex]\[
\text{The sum is a binomial with a degree of 6.}
\][/tex]
