Solve {l}{-x^{5}-7x^{4}-2x^{3}+x^{2}+4x+9}{x^5+7x^4+2x^3+2x^2-3x-9}

$$\left. \begin{array} { l } { - x ^ { 5 } - 7 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } + 4 x + 9 } \\ { x ^ { 5 } + 7 x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 3 x - 9 } \end{array} \right.$$

Least Common Multiple

$$\left(x^{2}-1\right)\left(x^{8}+14x^{7}+54x^{6}+43x^{5}+58x^{4}-22x^{3}-84x^{2}-63x-81\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(9+3x+7x^{2}+x^{3}\right)\left(x^{2}-1\right)=\left(x-1\right)\left(x+1\right)\left(x^{3}+7x^{2}+3x+9\right)$$

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

$$\left(x-1\right)\left(x+1\right)\left(x^{8}+14x^{7}+54x^{6}+43x^{5}+58x^{4}-22x^{3}-84x^{2}-63x-81\right)$$

Expand the expression.

$$x^{10}+14x^{9}+53x^{8}+29x^{7}+4x^{6}-65x^{5}-142x^{4}-41x^{3}+3x^{2}+63x+81$$

Evaluate

$$9+4x+x^{2}-2x^{3}-7x^{4}-x^{5},\ \left(x^{2}-1\right)\left(x^{3}+7x^{2}+3x+9\right)$$