To determine the nature of the completely simplified difference between the polynomials [tex]\( P_1 = a^3 b + 9 a^2 b^2 - 4 a b^5 \)[/tex] and [tex]\( P_2 = a^3 b - 3 a^2 b^2 + a b^5 \)[/tex], follow these steps:
1. Express the polynomials:
- [tex]\( P_1 = a^3 b + 9 a^2 b^2 - 4 a b^5 \)[/tex]
- [tex]\( P_2 = a^3 b - 3 a^2 b^2 + a b^5 \)[/tex]
2. Calculate the difference [tex]\( P_1 - P_2 \)[/tex]:
[tex]\[
P_{\text{diff}} = (a^3 b + 9 a^2 b^2 - 4 a b^5) - (a^3 b - 3 a^2 b^2 + a b^5)
\][/tex]
3. Distribute the negative sign in [tex]\( P_2 \)[/tex]:
[tex]\[
P_{\text{diff}} = a^3 b + 9 a^2 b^2 - 4 a b^5 - a^3 b + 3 a^2 b^2 - a b^5
\][/tex]
4. Combine like terms:
[tex]\[
P_{\text{diff}} = (a^3 b - a^3 b) + (9 a^2 b^2 + 3 a^2 b^2) + (-4 a b^5 - a b^5)
\][/tex]
Simplify each group:
[tex]\[
P_{\text{diff}} = 0 a^3 b + 12 a^2 b^2 - 5 a b^5
\][/tex]
Therefore:
[tex]\[
P_{\text{diff}} = 12 a^2 b^2 - 5 a b^5
\][/tex]
5. Identify the type class of the simplified difference:
- The result [tex]\( 12 a^2 b^2 - 5 a b^5 \)[/tex] is a binomial (since it has two terms).
- The degree of a term in a polynomial is the sum of the exponents of the variables in that 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].
- The degree of the polynomial is the highest degree among its terms, which is 6 in this case.
Hence, the completely simplified difference of the polynomials is a binomial with a degree of 6.
Answer:
The difference is a binomial with a degree of 6.
