Question 47: Factoring: \( {x^3} – x + 3{x^2}y + 3x{y^2} + {y^3} – y\)

\(\begin{array}{l} {x^3} – x + 3{x^2}y + 3x{y^2} + {y^3}–y\\ = \left( {{x) ^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) – \left( {x + y} \right) = {\left( {x + y} \right)^3} – \left( {x + y} \right)\\ = \left( {x + y} \right)\left[ {{{\left( {x + y} \right)}^2} – 1} \right] = \left( {x + y} \right)\left( {x + y + 1} \right)\left( {x + y – 1} \right) \end{array}\)