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