Question

Factorise the following: $27-125a^3-135a+225a^2$

Answer 1

The polynomial expression $27-125a^3-135a+225a^2$,

can be written as $3^3-{\left(5a\right)}^3-3{\left(3\right)}^2\left(5a\right)+3\left(3\right){\left(5a\right)}^2$ ….[1]

which is in the form of the algebraic identity ${\left(a-b\right)}^3=a^3-b^3-3ab\left(a-b\right)$

$=a^3-b^3-3a^{2}b+3a{b}^2$ ….[2]

By comparing equations [1] and [2], we can say that,

$27-125a^3-135a+225a^2={\left(3-5a\right)}^3$

Hence, $\mathbf{27}\mathbf{-}\mathbf{125}{\mathit{a}}^{\mathbf{3}}\mathbf{-}\mathbf{135}\mathit{a}\mathbf{+}\mathbf{225}{\mathit{a}}^{\mathbf{2}}\mathbf{=}\left(3-5a\right)\left(3-5a\right)\left(3-5a\right)$