For these sorts of factorisations involving rotation of variables, it would be a good idea to rearrange regarding only a specific variable. Example 1 Factorise \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \). \( \begin{align} &= (ac^2-b^c)+(a^2c-b^2c)+(a^3-b^3) \\ &= (a-b)c^2+(a^2-b^2)c+(a^3-b^3) \\ &= (a-b)c^2+(a-b)(a+b)c+(a-b)(a^2+ab+b^2) \\ &= (a-b)\left[ c^2+(a+b)c+(a^2+ab+b^2) \right] \\ &= (a-b)(a^2+b^2+c^2+ab+bc+ca) \end{align} \) Example 2 Factorise \( x^3+3ax^2+(3a^2-b^2)x+a^3-ab^2 \). \( […]
