Question

Factorise: a⁷ − ab⁶

Answer 1

Factorise the expression a⁷ − ab⁶ as shown below:
$$\begin{gathered} a^7-a{b}^6=a\left(a^6-b^6\right) \\ =a\left[{\left(a^3\right)}^2-{\left(b^3\right)}^2\right] \\ =a\left(a^3+b^3\right)\left(a^3-b^3\right)\left[\because \left(x-y\right)\left(x+y\right)=x^2-y^2\right] \end{gathered}$$
Now, using the identities $x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)\mathrm{and}{x}^3-y^3=\left(x-y\right)\left(x^2+y^2+xy\right)$, we get
$$\begin{gathered} a^7-a{b}^6=a\left(a^3+b^3\right)\left(a^3-b^3\right) \\ =a\left(a+b\right)\left(a^2+b^2-ab\right)\left(a-b\right)\left(a^2+b^2+ab\right) \\ =a\left(a-b\right)\left(a+b\right)\left(a^2+b^2-ab\right)\left(a^2+b^2+ab\right) \end{gathered}$$
Hence, $a^7-a{b}^6=a\left(a-b\right)\left(a+b\right)\left(a^2+b^2-ab\right)\left(a^2+b^2+ab\right)$.