Simplify the expression (2a^2*b - 3a*b^2 + b^2) - (a^2*b - 2a*b^2 + 2b).

Step-by-step solution

1. 1

   Distribute the minus sign across the whole second bracket. Subtracting a polynomial means adding its opposite, so every sign inside the second bracket flips, including the sign of the middle term which becomes $+2ab^{2}$: $2a^{2}b-3ab^{2}+b^{2}-a^{2}b+2ab^{2}-2b .$

2. 2

   Group terms by their variable part. Like terms must match in both variables and both exponents, so $a^{2}b$ pairs only with $a^{2}b$, and $ab^{2}$ only with $ab^{2}$: $\left(2a^{2}b-a^{2}b\right)+\left(-3ab^{2}+2ab^{2}\right)+b^{2}-2b .$

3. 3

   Combine each group. $2a^{2}b-a^{2}b=a^{2}b$ and $-3ab^{2}+2ab^{2}=-ab^{2}$.

4. 4

   Identify the terms with no partner. $b^{2}$ and $-2b$ appear only once each and cannot be combined with anything, since $b^{2}$ and $b$ are different powers.

5. 5

   Write the answer and check with numbers. The simplified expression is $a^{2}b-ab^{2}+b^{2}-2b$. At $a=2$, $b=1$: the original is $(8-6+1)-(4-4+2)=3-2=1$, and the answer gives $4-2+1-2=1$.