Solve {l}{x^{4}+x^{2}y^2+y^4}{x^4+4x^2+16}

$$\left. \begin{array} { l } { x ^ { 4 } + x ^ { 2 } y ^ { 2 } + y ^ { 4 } } \\ { x ^ { 4 } + 4 x ^ { 2 } + 16 } \end{array} \right.$$

Least Common Multiple

$$x^{8}+4x^{6}+16x^{4}+4x^{2}y^{4}+16y^{4}+y^{2}x^{6}+4y^{2}x^{4}+\left(xy\right)^{4}+16\left(xy\right)^{2}$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{4}+4x^{2}+16=\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)$$

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

$$x^{8}+4x^{6}+x^{4}y^{4}+16x^{4}+4x^{2}y^{4}+16x^{2}y^{2}+16y^{4}+y^{2}x^{6}+4y^{2}x^{4}$$

Evaluate

$$x^{4}+y^{4}+\left(xy\right)^{2},\ x^{4}+4x^{2}+16$$