Question

Factorise the given polynomial:

$3{\mathrm{x}}^3\mathrm{y}-243{\mathrm{xy}}^3$

Answer 1

Factorisation by using algebraic identities:

Consider the given polynomial $3{\mathrm{x}}^3\mathrm{y}-243{\mathrm{xy}}^3$,

$$\begin{gathered} \because 3{\mathrm{x}}^3\mathrm{y}-243{\mathrm{xy}}^3=3\mathrm{xy}\left(x^2-81y^2\right) \\ =3\mathrm{xy}\left(x^2-9^2{y}^2\right) \\ =3\mathrm{xy}\left(x^2-{\left(9y\right)}^2\right) \\ =3\mathrm{xy}\left(x-9y\right)\left(x+9y\right)[\because (a^2-b^2)=\left(a-b\right)\left(a+b\right)] \end{gathered}$$

Hence, $3{\mathrm{x}}^3\mathrm{y}-243{\mathrm{xy}}^3$$=$$3\mathrm{xy}\left(x-9y\right)\left(x+9y\right)$.