Answered

Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

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.

Sagot :

GinnyAnswer
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.