To determine the nature and degree of the difference between the given polynomials, we need to follow these steps:
1. Identify the terms of each polynomial:
- The first polynomial is [tex]\( a^3 b + 9 a^2 b^2 - 4 a b^5 \)[/tex].
- The second polynomial is [tex]\( a^3 b - 3 a^2 b^2 + a b^5 \)[/tex].
2. Subtract the corresponding terms of the second polynomial from the first polynomial:
- For the term [tex]\( a^3 b \)[/tex]:
[tex]\[
a^3 b - a^3 b = 0
\][/tex]
- For the term [tex]\( 9 a^2 b^2 \)[/tex] and [tex]\( -3 a^2 b^2 \)[/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]
- For the term [tex]\( -4 a b^5 \)[/tex] and [tex]\( a b^5 \)[/tex]:
[tex]\[
-4 a b^5 - a b^5 = -4 a b^5 - 1 a b^5 = -5 a b^5
\][/tex]
3. Combine the result of the subtraction:
[tex]\[
0 + 12 a^2 b^2 - 5 a b^5
\][/tex]
4. Analyze the resulting polynomial:
- The resulting polynomial is [tex]\( 12 a^2 b^2 - 5 a b^5 \)[/tex].
- This polynomial has two non-zero terms: [tex]\( 12 a^2 b^2 \)[/tex] and [tex]\( -5 a b^5 \)[/tex].
5. Identify the type and degree of the polynomial:
- Number of terms: The polynomial has two non-zero terms, which classifies it as a binomial.
- Degree: The degree of a term is the sum of the exponents of the variables:
- The degree of [tex]\( 12 a^2 b^2 \)[/tex] is [tex]\( 2 + 2 = 4 \)[/tex].
- The degree of [tex]\( -5 a b^5 \)[/tex] is [tex]\( 1 + 5 = 6 \)[/tex].
- The highest degree among the terms (4 and 6) is 6; thus, the degree of the polynomial is 6.
Hence, the correct classification is:
The difference is a binomial with a degree of 6.
