Question

Factorize:
x⁶ − 729

Answer 1

$$\begin{gathered} x^6-729={\left(x^2\right)}^3-{\left(9\right)}^3 \\ =\left[x^2-9\right]\left[{\left(x^2\right)}^2+x^2\times 9+9^2\right] \\ =\left[x^2-3^2\right]\left(x^4+9x^2+81\right) \\ =\left(x+3\right)\left(x-3\right)\left(x^4+18x^2+81-9x^2\right) \\ =\left(x+3\right)\left(x-3\right)\left[{\left(x^2\right)}^2+2\times x^2\times 9+9^2-9x^2\right] \\ =\left(x+3\right)\left(x-3\right)\left[{\left(x^2+9\right)}^2-{\left(3x\right)}^2\right] \\ =\left(x+3\right)\left(x-3\right)\left(x^2+9+3x\right)\left(x^2+9-3x\right) \\ =\left(x+3\right)\left(x-3\right)\left(x^2+3x+9\right)\left(x^2-3x+9\right) \end{gathered}$$