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

Factorization

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

Example 1

$$\textup{Factorize } 8x^3+27 $$

Solution

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

Example 2

$$\textup{Factorize } 27x^3+64 $$

Solution

\begin{align*} & \because a^3+b^3=(a+b)(a^2-ab+b^2)\\\\ &\therefore\;\; 27x^3+64\\ &=(3x)^3+(4)^3\\ &=(3x+4)[(3x)^2-(3x)(4)+(4)^2]\\ &=(3x+4)(9x^2-12x+16)\\ \end{align*}

Example 3

$$\textup{Factorize } y^3+125 $$

Solution

\begin{align*} &\because a^3+b^3=(a+b)(a^2-ab+b^2)\\\\ &\therefore\;\; y^3+125\\ &=(y)^3+(5)^3\\ &=(y+5)[(y)^2-(y)(5)+(5)^2]\\ &=(y+5)(y^2-5y+25)\\ \end{align*}

Example 4

$$\textup{Factorize } 8x^3+1 $$

Solution

\begin{align*} &\because a^3+b^3=(a+b)(a^2-ab+b^2)\\\\ &\therefore\;\; 8x^3+1\\ &=(2x)^3+(1)^3\\ &=(2x+1)[(2x)^2-(2x)(1)+(1)^2]\\ &=(2x+1)(4x^2-2x+1)\\ \end{align*}