Algebraic Identity (a-b)(a2+ab+b2)=a3-b3

Algebraic Identity

(a-b)(a2+ab+b2)=a3-b3

Example 1

$$\textup{Expand } (x-3)(x^2+3x+9)$$

Solution

\begin{align*} &\because(a-b)(a^2+ab+b^2)=a^3-b^3\\\\ &\therefore(x-3)(x^2+3x+9)\\ &=(x-3)[(x)^2+(x)(3)+(3)^2]\\ &=(x)^3-(3)^3\\ &=x^3-27\\ \end{align*}

Example 2

$$\textup{Expand } (c-2)(c^2+2c+4)$$

Solution

\begin{align*} &\because (a-b)(a^2+ab+b^2)=a^3-b^3\\\\ &\therefore (c-2)(c^2+2c+4)\\ &=(c-2)[(c)^2+(2)(2)+(2)^2]\\ &=(c)^3-(2)^3\\ &=c^3-8\\ \end{align*}

Example 3

$$\textup{Expand } (2x-3)(4x^2+6x+9)$$

Solution

\begin{align*} &\because(a-b)(a^2+ab+b^2)=a^3-b^3\\\\ &\therefore (2x-3)(4x^2+6x+9)\\ &=(2x-3)[(2x)^2+(2x)(3)+(3)^2]\\ &=(2x)^3-(3)^3\\ &=8x^3-27\\ \end{align*}

Example 4

$$\textup{Expand } (y-4)(y^2+4y+16)$$

Solution

\begin{align*} &\because(a-b)(a^2+ab+b^2)=a^3-b^3\\\\ &\therefore(y-4)(y^2+4y+16)\\ &=(y-4)[(y)^2+(y)(4)+(4)^2]\\ &=(y)^3-(4)^3\\ &=y^3-64\\ \end{align*}