Question

Factorize: $27-125a^3-135a+225a^2$.

Answer 1

Factorize the given polynomial

Given: $27-125a^3-135a+225a^2$

$$\begin{gathered} 27-125a^3-135a+225a^2 \\ =-1(125a^3-27-225a^2+135a) \\ =-1\left[{\left(5a\right)}^3-{\left(3\right)}^3-3\times {\left(5a\right)}^2\times 3+3\times 5a\times 3^2\right] \end{gathered}$$

using the identity $a^3-b^3-3a^{2}b+3a{b}^2={(a-b)}^3$

$$\begin{gathered} 27-125a^3-135a+225a^2=-1{(5a-3)}^3 \\ ={(3-5a)}^3 \\ =(3-5a)(3-5a)(3-5a) \end{gathered}$$

Hence, the factorized form of the polynomial $27-125a^3-135a+225a^2$ is $(3-5a)(3-5a)(3-5a)$.