Solve {l}{(x^2+x+1)(x^3+1)}{(x^2+x+1)(x+1)(x^2-x+1)}

$$\left. \begin{array} { l } { ( x ^ { 2 } + x + 1 ) ( x ^ { 3 } + 1 ) } \\ { ( x ^ { 2 } + x + 1 ) ( x + 1 ) ( x ^ { 2 } - x + 1 ) } \end{array} \right.$$

Least Common Multiple

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

Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$x^{5}+x^{4}+x^{3}+x^{2}+x+1$$

Evaluate

$$\left(x+1\right)\left(x^{4}+x^{2}+1\right),\ \left(x+1\right)\left(x^{4}+x^{2}+1\right)$$