Algebraic Identity

Example 1

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

Solution

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

Example 2

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

Solution

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

Example 3

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

Solution

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

Example 4

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

Solution

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