Solve {l}{x^{4}+4}{x^4-18x^2+1}{x^4-14x^2+25}

$$\left. \begin{array} { l } { x ^ { 4 } + 4 } \\ { x ^ { 4 } - 18 x ^ { 2 } + 1 } \\ { x ^ { 4 } - 14 x ^ { 2 } + 25 } \end{array} \right.$$

Least Common Multiple

$$\left(x^{4}+4\right)\left(x^{8}-32x^{6}+278x^{4}-464x^{2}+25\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{4}+4=\left(x^{2}-2x+2\right)\left(x^{2}+2x+2\right) x^{4}-18x^{2}+1=\left(x^{2}-\left(-4\sqrt{5}+9\right)\right)\left(x^{2}-\left(4\sqrt{5}+9\right)\right) x^{4}-14x^{2}+25=\left(x^{2}-\left(-2\sqrt{6}+7\right)\right)\left(x^{2}-\left(2\sqrt{6}+7\right)\right)$$

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

$$\left(x^{4}+4\right)\left(x^{8}-32x^{6}+278x^{4}-464x^{2}+25\right)$$

Expand the expression.

$$x^{12}-32x^{10}+282x^{8}-592x^{6}+1137x^{4}-1856x^{2}+100$$

Evaluate

$$x^{4}+4,\ x^{4}-18x^{2}+1,\ x^{4}-14x^{2}+25$$