Let’s solve the problem step by step.
We have two polynomials:
[tex]\[ P_1(a, b) = a^3 b + 9 a^2 b^2 - 4 a b^5 \][/tex]
[tex]\[ P_2(a, b) = a^3 b - 3 a^2 b^2 + a b^5 \][/tex]
First, compute the difference between these two polynomials:
[tex]\[ P_1(a, b) - P_2(a, b) \][/tex]
Simplify the subtraction term by term:
1. For the term [tex]\(a^3 b\)[/tex]:
[tex]\[ a^3 b - a^3 b = 0 \][/tex]
2. For the term [tex]\(9 a^2 b^2\)[/tex]:
[tex]\[ 9 a^2 b^2 - (-3 a^2 b^2) = 9 a^2 b^2 + 3 a^2 b^2 = 12 a^2 b^2 \][/tex]
3. For the term [tex]\(-4 a b^5\)[/tex]:
[tex]\[ -4 a b^5 - a b^5 = -4 a b^5 - 1 a b^5 = -5 a b^5 \][/tex]
Thus, the simplified polynomial difference is:
[tex]\[ P_1(a, b) - P_2(a, b) = 0 + 12 a^2 b^2 - 5 a b^5 \][/tex]
[tex]\[ P_1(a, b) - P_2(a, b) = 12 a^2 b^2 - 5 a b^5 \][/tex]
Now, let’s analyze the resulting polynomial [tex]\(12 a^2 b^2 - 5 a b^5\)[/tex]:
- The polynomial has 2 terms.
- The degree of a term is the sum of the exponents of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. 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].
Since the term [tex]\(-5 a b^5\)[/tex] has the highest degree, the resulting polynomial has a degree of 6.
To summarize:
- The difference is a binomial (since it has 2 terms).
- The degree of the difference is 6.
Therefore, the correct and true statement is:
The difference is a binomial with a degree of 6.
