Solve {l}{(4-x^{3})(16+4x^{3}+x^{3})}{(4-x^{3})(16+4x^{3}+x^{6})}{(4-x^{5})(16+4x^{5}+x^5)}{(4-x^5)(16+4x^5+x^10)}

$$\left. \begin{array} { l } { ( 4 - x ^ { 3 } ) ( 16 + 4 x ^ { 3 } + x ^ { 3 } ) } \\ { ( 4 - x ^ { 3 } ) ( 16 + 4 x ^ { 3 } + x ^ { 6 } ) } \\ { ( 4 - x ^ { 5 } ) ( 16 + 4 x ^ { 5 } + x ^ { 5 } ) } \\ { ( 4 - x ^ { 5 } ) ( 16 + 4 x ^ { 5 } + x ^ { 10 } ) } \end{array} \right.$$

Least Common Multiple

$$lcm(64+4x^{3}-5x^{6},64-x^{9},64+4x^{5}-5x^{10},64-x^{15})$$

Solution Steps

Factor the expressions that are not already factored.

$$64+4x^{3}-5x^{6}=\left(-x^{3}+4\right)\left(5x^{3}+16\right) 64-x^{9}=\left(x^{3}-4\right)\left(-x^{6}-4x^{3}-16\right) 64+4x^{5}-5x^{10}=\left(-x^{5}+4\right)\left(5x^{5}+16\right) 64-x^{15}=\left(x^{5}-4\right)\left(-x^{10}-4x^{5}-16\right)$$

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

$$25x^{37}+80x^{34}-20x^{32}-64x^{29}-1600x^{28}-320x^{27}-5120x^{25}-1024x^{24}+1280x^{23}-1600x^{22}+4096x^{20}-5120x^{19}+20480x^{18}+1280x^{17}+65536x^{15}+4096x^{14}+102400x^{13}+20480x^{12}+327680x^{10}+65536x^{9}-81920x^{8}-262144x^{5}-1310720x^{3}-4194304$$

Evaluate

$$64+4x^{3}-5x^{6},\ 64-x^{9},\ 64+4x^{5}-5x^{10},\ 64-x^{15}$$