To determine the product of the expressions [tex]\(\left(-6 a^3 b + 2 a b^2\right)\left(5 a^2 - 2 a b^2 - b\right)\)[/tex], let's consider each term in the first polynomial and multiply it by each term in the second polynomial.
The first polynomial can be represented as:
[tex]\[ P_1 = -6a^3b + 2ab^2 \][/tex]
The second polynomial can be represented as:
[tex]\[ P_2 = 5a^2 - 2ab^2 - b \][/tex]
Now we multiply each term from [tex]\( P_1 \)[/tex] with every term from [tex]\( P_2 \)[/tex]:
### Step-by-Step Multiplication:
1. Multiply [tex]\(-6a^3b\)[/tex] by each term in [tex]\( P_2 \)[/tex]:
[tex]\[ -6a^3b \cdot 5a^2 = -30a^5b \][/tex]
[tex]\[ -6a^3b \cdot (-2ab^2) = 12a^4b^3 \][/tex]
[tex]\[ -6a^3b \cdot (-b) = 6a^3b^2 \][/tex]
2. Multiply [tex]\( 2ab^2 \)[/tex] by each term in [tex]\( P_2 \)[/tex]:
[tex]\[ 2ab^2 \cdot 5a^2 = 10a^3b^2 \][/tex]
[tex]\[ 2ab^2 \cdot (-2ab^2) = -4a^2b^4 \][/tex]
[tex]\[ 2ab^2 \cdot (-b) = -2ab^3 \][/tex]
### Collecting and Combining Like Terms:
Now, let's list all the resulting terms from our multiplications:
[tex]\[ -30a^5b, \, 12a^4b^3, \, 6a^3b^2, \, 10a^3b^2, \, -4a^2b^4, \, -2ab^3 \][/tex]
Combining like terms:
- For [tex]\( a^5b \)[/tex]:
[tex]\[ -30a^5b \][/tex]
- For [tex]\( a^4b^3 \)[/tex]:
[tex]\[ 12a^4b^3 \][/tex]
- For [tex]\( a^3b^2 \)[/tex]:
[tex]\[ 6a^3b^2 + 10a^3b^2 = 16a^3b^2 \][/tex]
- For [tex]\( a^2b^4 \)[/tex]:
[tex]\[ -4a^2b^4 \][/tex]
- For [tex]\( ab^3 \)[/tex]:
[tex]\[ -2ab^3 \][/tex]
### Final Product:
Thus, the final product of the multiplication of the given polynomials is:
[tex]\[ -30a^5b + 12a^4b^3 + 16a^3b^2 - 4a^2b^4 - 2ab^3 \][/tex]
So, the product is:
[tex]\[
-30 a^5 b + 12 a^4 b^3 + 16 a^3 b^2 - 4 a^2 b^4 - 2 ab^3
\][/tex]
