Solve {l}{(4x^{3}-7x+2)(3x^2-4x+6)}{(x^6-12x^3+32)div(x^3-4)}

$$\left. \begin{array} { l } { ( 4 x ^ { 3 } - 7 x + 2 ) ( 3 x ^ { 2 } - 4 x + 6 ) } \\ { ( x ^ { 6 } - 12 x ^ { 3 } + 32 ) \div ( x ^ { 3 } - 4 ) } \end{array} \right.$$

Least Common Multiple

$$\left(3x^{2}-4x+6\right)\left(x^{3}-8\right)\left(4x^{3}-7x+2\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$34x^{2}-50x-16x^{4}+12+12x^{5}+3x^{3}=\left(3x^{2}-4x+6\right)\left(4x^{3}-7x+2\right) x^{3}-8=\left(x-2\right)\left(x^{2}+2x+4\right)$$

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

$$\left(x-2\right)\left(3x^{2}-4x+6\right)\left(x^{2}+2x+4\right)\left(4x^{3}-7x+2\right)$$

Expand the expression.

$$12x^{8}-16x^{7}+3x^{6}-62x^{5}+78x^{4}-12x^{3}-272x^{2}+400x-96$$

Evaluate

$$\left(3x^{2}-4x+6\right)\left(4x^{3}-7x+2\right),\ x^{3}-8$$