Factorization a3-3a2b+3ab2-b3=(a-b)3
Factorization
a3-3a2b+3ab2-b3=(a-b)3
Example 1:
$$\textup{Factorize } x^3-9x^2+27x-27 $$
Solution:
\begin{align*}
&\because\;\; a^3-3a^2b+3ab^2-b^3=(a-b)^3\\\\
&\therefore\;\; x^3-9x^2+27x-27 \\
&=(x)^3-3(x)^2(3)+3(x)(3)^2-(3)^3\\
&=(x-3)^3\\
\end{align*}
Example 2:
$$\textup{Factorize } x^3-12x^2+48x-64 $$
Solution:
\begin{align*}
&\because\;\; a^3-3a^2b+3ab^2-b^3=(a-b)^3\\\\
&\therefore\;\; x^3-12x^2+48x-64 \\
&=(x)^3-3(x)^2(4)+3(x)(4)^2-(4)^3\\
&=(x-4)^3\\
\end{align*}
Example 3:
$$\textup{Factorize } y^3-6y^2+12y-8 $$
Solution:
\begin{align*}
&\because\;\; a^3-3a^2b+3ab^2-b^3=(a-b)^3\\\\
&\therefore\;\; y^3-6y^2+12y-8 \\
&=(y)^3-3(y)^2(2)+3(y)(2)^2-(2)^3\\
&=(y-2)^3\\
\end{align*}
Example 4:
$$\textup{Factorize } z^3-15z^2+75z-125 $$
Solution:
\begin{align*}
&\because\;\; a^3-3a^2b+3ab^2-b^3=(a-b)^3\\\\
&\therefore\;\; z^3-15z^2+75z-125 \\
&=(z)^3-3(z)^2(5)+3(z)(5)^2-(5)^3\\
&=(z-5)^3\\
\end{align*}