Solve {l}{9x^{6}+12x^{5}+4x}{x^{4}-2x^{3}+x^{2}}{3x^4-12x^2+2x}{x^4-8x^2+16}

$$\left. \begin{array} { l } { 9 x ^ { 6 } + 12 x ^ { 5 } + 4 x } \\ { x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } } \\ { 3 x ^ { 4 } - 12 x ^ { 2 } + 2 x } \\ { x ^ { 4 } - 8 x ^ { 2 } + 16 } \end{array} \right.$$

Least Common Multiple

$$\left(27x^{8}+36x^{7}-108x^{6}-126x^{5}+24x^{4}+12x^{3}-48x+8\right)\left(x\left(x-1\right)\left(x^{2}-4\right)\right)^{2}$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(x^{2}-4\right)^{2}=\left(x-2\right)^{2}\left(x+2\right)^{2}$$

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

$$x^{2}\left(x-1\right)^{2}\left(x^{2}-4\right)^{2}\left(27x^{8}+36x^{7}-108x^{6}-126x^{5}+24x^{4}+12x^{3}-48x+8\right)$$

Expand the expression.

$$27x^{16}-18x^{15}-369x^{14}+270x^{13}+1824x^{12}-1458x^{11}-3792x^{10}+3276x^{9}+2792x^{8}-2368x^{7}-824x^{6}-64x^{5}+1600x^{4}-1024x^{3}+128x^{2}$$

Evaluate

$$x\left(9x^{5}+12x^{4}+4\right),\ \left(x\left(x-1\right)\right)^{2},\ x\left(3x^{3}-12x+2\right),\ \left(x^{2}-4\right)^{2}$$