What is true about the completely simplified sum of the polynomials [tex]\(3x^2y^2 - 2xy^5\)[/tex] and [tex]\(-3x^2y^2 + 3x^4y\)[/tex]?

A. The sum is a trinomial with a degree of 5.
B. The sum is a trinomial with a degree of 6.
C. The sum is a binomial with a degree of 5.
D. The sum is a binomial with a degree of 6.



Answer :

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]