To determine the true statement about the completely simplified difference of the 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], let's follow these steps:
1. Write down the given polynomials:
- [tex]\( \text{poly1} = a^3 b + 9 a^2 b^2 - 4 a b^5 \)[/tex]
- [tex]\( \text{poly2} = a^3 b - 3 a^2 b^2 + a b^5 \)[/tex]
2. Calculate the difference of the polynomials:
[tex]\( \text{difference} = \text{poly1} - \text{poly2} \)[/tex]
Subtract each term of [tex]\(\text{poly2}\)[/tex] from the corresponding term in [tex]\(\text{poly1}\)[/tex]:
[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]
3. 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]
Simplifying each group:
[tex]\[
\text{difference} = 0 + 12 a^2 b^2 - 5 a b^5
\][/tex]
4. Simplify the expression:
[tex]\[
\text{difference} = 12 a^2 b^2 - 5 a b^5
\][/tex]
The simplified difference is [tex]\( 12 a^2 b^2 - 5 a b^5 \)[/tex].
5. Determine the number of terms and the degree:
- There are 2 terms in the simplified difference.
- The degree is the highest sum of the exponents in any term. The term [tex]\( -5 a b^5 \)[/tex] has a degree of [tex]\( 1 + 5 = 6 \)[/tex] and the term [tex]\( 12 a^2 b^2 \)[/tex] has a degree of [tex]\( 2 + 2 = 4 \)[/tex]. Therefore, the overall degree is the highest of these, which is 6.
6. Conclusion:
The correct statement about the completely simplified difference of the polynomials is:
- The difference is a binomial (because it has two terms) with a degree of 6.
Therefore, the correct answer is:
The difference is a binomial with a degree of 6.
