Solve {l}{2(a-b)^{2}-3(b-a)-2(b-a)^{2}+4(a-b)-(a-b)^{3}}{2b^{2}-3b+3a-2b^2+2a^2+4a-4b-a^3+b^3}

$$\left. \begin{array} { l } { 2 ( a - b ) ^ { 2 } - 3 ( b - a ) - 2 ( b - a ) ^ { 2 } + 4 ( a - b ) - ( a - b ) ^ { 3 } } \\ { 2 b ^ { 2 } - 3 b + 3 a - 2 b ^ { 2 } + 2 a ^ { 2 } + 4 a - 4 b - a ^ { 3 } + b ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$\left(a-b\right)\left(a^{5}-2a^{4}-14a^{3}-a^{2}b^{3}+14a^{2}+2ab^{4}-21ab^{2}+49a-b^{5}+14b^{3}+b^{2}a^{3}-2ba^{4}+4ba^{3}+21ba^{2}-49b-2\left(ab\right)^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$7a-7b-\left(a-b\right)^{3}=\left(a-b\right)\left(-a^{2}+2ab-b^{2}+7\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$\left(a-b\right)\left(a^{5}-2a^{4}-14a^{3}-a^{2}b^{3}-2a^{2}b^{2}+14a^{2}+2ab^{4}-21ab^{2}+49a-b^{5}+14b^{3}+b^{2}a^{3}-2ba^{4}+4ba^{3}+21ba^{2}-49b\right)$$

Expand the expression.

$$a^{6}-2a^{5}-14a^{4}-2a^{3}b^{3}+14a^{3}+3a^{2}b^{4}+2a^{2}b^{3}-42a^{2}b^{2}+49a^{2}-3ab^{5}+35ab^{3}-98ab+b^{6}-14b^{4}+3b^{2}a^{4}-6b^{2}a^{3}+49b^{2}-3ba^{5}+6ba^{4}+35ba^{3}-14ba^{2}$$

Evaluate

$$-\left(a-b\right)^{3}+7a-7b,\ -a^{3}+2a^{2}+7a+b^{3}-7b$$