Solve {l}{35m^{2}n^{3}-70m^3}{x^3+x^5-x^7}

$$\left. \begin{array} { l } { 35 m ^ { 2 } n ^ { 3 } - 70 m ^ { 3 } } \\ { x ^ { 3 } + x ^ { 5 } - x ^ { 7 } } \end{array} \right.$$

Least Common Multiple

$$35m^{2}x^{3}\left(2m-n^{3}\right)\left(1+x^{2}-x^{4}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{3}\left(1+x^{2}-x^{4}\right)=-\left(x^{2}-\left(-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\right)\left(x^{2}-\left(\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\right)x^{3}$$

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

$$35m^{2}x^{3}\left(2m-n^{3}\right)\left(1+x^{2}-x^{4}\right)$$

Expand the expression.

$$35m^{2}n^{3}x^{7}-70m^{3}x^{7}-35m^{2}n^{3}x^{5}+70m^{3}x^{5}-35m^{2}n^{3}x^{3}+70m^{3}x^{3}$$

Evaluate

$$35m^{2}\left(n^{3}-2m\right),\ x^{3}\left(1+x^{2}-x^{4}\right)$$