Question

Which of the following is a factor of (x + y)³ – (x³ + y³)?
(a) x² + 2xy + y²
(b) x² – xy + y²
(c) xy²
(d) 3xy

Answer 1

$$\begin{gathered} {\left(x+y\right)}^3-\left(x^3+y^3\right) \\ ={\left(x\right)}^3+{\left(y\right)}^3+3\left(x\right)\left(y\right)\left(x+y\right)-\left(x^3+y^3\right)\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:{\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right)\right) \\ =x^3+y^3+3xy\left(x+y\right)-x^3-y^3 \\ =3xy\left(x+y\right) \\ \mathrm{Thus},{\left(x+y\right)}^3-\left(x^3+y^3\right)\mathrm{has}\mathrm{two}\mathrm{factors}\left(3xy\right)\mathrm{and}\left(x+y\right). \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{d}\right). \end{gathered}$$