To solve this problem, we need to determine the difference between the two given polynomials [tex]\(a^3 b + 9 a^2 b^2 - 4 a b^5\)[/tex] and [tex]\(a^3 b - 3 a^2 b^2 + a b^5\)[/tex], and then simplify the result. We will then analyze the simplified polynomial to identify its number of terms (binomial, trinomial, etc.) and its degree.
Here are the primary steps involved:
### 1. Subtract the Polynomials
First, let's find the difference between the two polynomials:
[tex]\[
\text{difference} = (a^3 b + 9 a^2 b^2 - 4 a b^5) - (a^3 b - 3 a^2 b^2 + a b^5)
\][/tex]
Remove the parentheses:
[tex]\[
\text{difference} = a^3 b + 9 a^2 b^2 - 4 a b^5 - a^3 b + 3 a^2 b^2 - a b^5
\][/tex]
### 2. Combine Like Terms
Next, we combine like terms:
[tex]\[
\text{difference} = [(a^3 b - a^3 b)] + [(9 a^2 b^2 + 3 a^2 b^2)] + [(-4 a b^5 - a b^5)]
\][/tex]
Simplify each group of like terms:
[tex]\[
\text{difference} = 0 a^3 b + 12 a^2 b^2 - 5 a b^5
\][/tex]
So, the simplified difference is:
[tex]\[
12 a^2 b^2 - 5 a b^5
\][/tex]
### 3. Determine the Number of Terms and the Degree
We now examine the simplified polynomial [tex]\(12 a^2 b^2 - 5 a b^5\)[/tex]:
- Number of terms: This polynomial has two terms.
- Type: Since it has two terms, it is a binomial.
- Degree: To find the degree of the polynomial, we look at the exponents of each term. For [tex]\(12 a^2 b^2\)[/tex], the degree is [tex]\(2 + 2 = 4\)[/tex]. For [tex]\(-5 a b^5\)[/tex], the degree is [tex]\(1 + 5 = 6\)[/tex]. Therefore, the degree of the polynomial is the highest degree among its terms, which is 6.
### Conclusion
Given the simplified polynomial has two terms (making it a binomial) and the highest degree term is 6, we conclude that:
- The difference is a binomial with a degree of 6.
Thus, the correct statement about the completely simplified difference of the polynomials is:
The difference is a binomial with a degree of 6.
