To divide [tex]\( \left(a^3 - a^2 b - a b + b^2\right) \)[/tex] by [tex]\( (a - b) \)[/tex], let's perform polynomial long division step-by-step:
1. Write down the dividend and divisor:
[tex]\[
\text{Dividend} = a^3 - a^2 b - a b + b^2
\][/tex]
[tex]\[
\text{Divisor} = a - b
\][/tex]
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\text{First Term: } \frac{a^3}{a} = a^2
\][/tex]
3. Multiply the entire divisor by this term and subtract from the dividend:
[tex]\[
(a - b) \times a^2 = a^3 - a^2 b
\][/tex]
[tex]\[
\text{New Dividend: } (a^3 - a^2 b - a b + b^2) - (a^3 - a^2 b) = 0 - a b + b^2 = -a b + b^2
\][/tex]
4. Repeat the process with the new dividend:
Now, divide the new dividend's first term by the first term of the divisor:
[tex]\[
\text{Next Term: } \frac{-a b}{a} = -b
\][/tex]
Multiply the entire divisor by this term and subtract from the new dividend:
[tex]\[
(a - b) \times -b = -a b + b^2
\][/tex]
[tex]\[
\text{Next New Dividend: } (-a b + b^2) - (-a b + b^2) = 0
\][/tex]
5. Combine the terms obtained from the steps above:
The terms we obtained are [tex]\( a^2 \)[/tex] and [tex]\( -b \)[/tex].
Therefore, the result of dividing [tex]\( a^3 - a^2 b - a b + b^2 \)[/tex] by [tex]\( a - b \)[/tex] is:
[tex]\[
a^2 - b
\][/tex]
[tex]\(\boxed{a^2 - b}\)[/tex]
