Solve {l}{x^{4}-4x^{3}-x^{2}+16x-12}{x^5-3x^4-5x^3+27x^2-32x+1}

$$\left. \begin{array} { l } { x ^ { 4 } - 4 x ^ { 3 } - x ^ { 2 } + 16 x - 12 } \\ { x ^ { 5 } - 3 x ^ { 4 } - 5 x ^ { 3 } + 27 x ^ { 2 } - 32 x + 1 } \end{array} \right.$$

Least Common Multiple

$$\left(x-3\right)\left(x-1\right)\left(x^{2}-4\right)\left(x^{5}-3x^{4}-5x^{3}+27x^{2}-32x+1\right)$$

Solution Steps

Factor the expressions that are not already factored.

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

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

$$\left(x-3\right)\left(x-2\right)\left(x-1\right)\left(x+2\right)\left(x^{5}-3x^{4}-5x^{3}+27x^{2}-32x+1\right)$$

Expand the expression.

$$x^{9}-7x^{8}+6x^{7}+66x^{6}-195x^{5}+58x^{4}+520x^{3}-837x^{2}+400x-12$$

Evaluate

$$\left(x-3\right)\left(x-1\right)\left(x^{2}-4\right),\ x^{5}-3x^{4}-5x^{3}+27x^{2}-32x+1$$