Algebraic Identity (a+b)3=a3+3a2b+3ab2+b3
Algebraic Identity
(a+b)3=a3+3a2b+3ab2+b3
Example 1
$$\textup{Expand } (x+3)^3 $$
Solution
\begin{align*}
\because\;\;(a+b)^3&=a^3+3a^2b+3ab^2+b^3\\\\
\therefore\;\; (x+3)^3&=(x)^3+3(x)^2(3)+3(x)(3)^2+(3)^3\\
&=x^3+3(x^2)(3)+3(x)(9)+27\\
&=x^3+9x^2+27x+27\\
\end{align*}
Example 2
$$\textup{Expand } (x+5)^3 $$
Solution
\begin{align*}
\because\;\;(a+b)^3&=a^3+3a^2b+3ab^2+b^3\\\\
\therefore\;\; (x+5)^3&=(x)^3+3(x)^2(5)+3(x)(5)^2+(5)^3\\
&=x^3+3(x^2)(5)+3(x)(25)+125\\
&=x^3+15x^2+75x+125\\
\end{align*}
Example 3
$$\textup{Expand } (2b+c)^3 $$
Solution
\begin{align*}
\because\;\;(a+b)^3&=a^3+3a^2b+3ab^2+b^3\\\\
\therefore\;\; (2b+c)^3&=(2b)^3+3(2b)^2(c)+3(2b)(c)^2+(c)^3\\
&=8b^3+3(4b^2)(c)+3(2b)(c^2)+c^3\\
&=8b^3+12b^2c+6bc^2+c^3\\
\end{align*}
Example 4
$$\textup{Expand } (c+3d)^3 $$
Solution
\begin{align*}
\because\;\;(a+b)^3&=a^3+3a^2b+3ab^2+b^3\\\\
\therefore\;\; (c+3d)^3&=(c)^3+3(c)^2(3d)+3(c)(3d)^2+(3d)^3\\
&=c^3+3(c^2)(3d)+3(c)(9d^2)+27d^3\\
&=c^3+9c^2d+27cd^2+27d^3\\
\end{align*}