Solve {l}{2ab^{3}x+2b^{4}x^{3}+10a^{2}b^{4}x^{2}-10ab}{8ab^{3}x+12b^{4}x^3+10a^2b^4x^2}

$$\left. \begin{array} { l } { 2 a b ^ { 3 } x + 2 b ^ { 4 } x ^ { 3 } + 10 a ^ { 2 } b ^ { 4 } x ^ { 2 } - 10 a b } \\ { 8 a b ^ { 3 } x + 12 b ^ { 4 } x ^ { 3 } + 10 a ^ { 2 } b ^ { 4 } x ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$2xb^{3}\left(6b^{4}x^{5}+25x^{3}\left(ab\right)^{4}+25x^{2}\left(ab\right)^{3}-30abx^{2}+4x\left(ab\right)^{2}-25bxa^{3}+35a^{2}\left(bx\right)^{4}+10a\left(bx\right)^{3}-20a^{2}\right)$$

Solution Steps

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

$$2xb^{3}\left(6b^{4}x^{5}+35a^{2}b^{4}x^{4}+25x^{3}a^{4}b^{4}+10ab^{3}x^{3}+25x^{2}a^{3}b^{3}-30abx^{2}+4xa^{2}b^{2}-25bxa^{3}-20a^{2}\right)$$

Expand the expression.

$$12x^{6}b^{7}+70a^{2}x^{5}b^{7}+50a^{4}x^{4}b^{7}+20ax^{4}b^{6}+50a^{3}x^{3}b^{6}-60ax^{3}b^{4}+8a^{2}x^{2}b^{5}-50x^{2}a^{3}b^{4}-40xa^{2}b^{3}$$

Evaluate

$$2b\left(axb^{2}+5b^{3}\left(ax\right)^{2}+\left(bx\right)^{3}-5a\right),\ 2xb^{3}\left(6bx^{2}+5bxa^{2}+4a\right)$$