Algebraic Identity (a-b)3=a3-3a2b+3ab2-b3
Algebraic Identity
(a-b)3=a3-3a2b+3ab2-b3
Example 1
$$\textup{Expand } (x-2)^3 $$
Solution
\begin{align*}
\because\;\;(a-b)^3&=a^3-3a^2b+3ab^2-b^3\\\\
\therefore\;\; (x-2)^3&=(x)^3-3(x)^2(2)+3(x)(2)^2-(2)^3\\
&=x^3-3(x^2)(2)+3(x)(4)-8\\
&=x^3-6x^2+12x-8\\
\end{align*}
Example 2
$$\textup{Expand } (y-1)^3 $$
Solution
\begin{align*}
\because\;\;(a-b)^3&=a^3-3a^2b+3ab^2-b^3\\\\
\therefore\;\; (y-1)^3&=(y)^3-3(y)^2(1)+3(y)(1)^2-(1)^3\\
&=y^3-3(y^2)(1)+3(y)(1)-1\\
&=y^3-3y^2+3y-1\\
\end{align*}
Example 3
$$\textup{Expand } (2x-3)^3 $$
Solution
\begin{align*}
\because\;\;(a-b)^3&=a^3-3a^2b+3ab^2-b^3\\\\
\therefore\;\; (2x-3)^3&=(2x)^3-3(2x)^2(3)+3(2x)(3)^2-(3)^3\\
&=8x^3-3(4x^2)(3)+3(2x)(9)-27\\
&=8x^3-36x^2+54x-27\\
\end{align*}
Example 4
$$\textup{Expand } (y-2z)^3 $$
Solution
\begin{align*}
\because\;\;(a-b)^3&=a^3-3a^2b+3ab^2+b^3\\\\
\therefore\;\; (y-2z)^3&=(y)^3-3(y)^2(2z)+3(y)(2z)^2-(2z)^3\\
&=y^3-3(y^2)(2z)+3(y)(4z^2)-8z^3\\
&=y^3-6y^2z+12yz^2-8z^2\\
\end{align*}