Cubic Factorization
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
Example 1
Factorize \(x^3 + 8\)
\begin{equation*} \begin{split} & x^3 + 8 \\\\ & x^3 + 2^3\\\\ & (x + 2)(x^2 + 2x + 4) \end{split} \end{equation*}
Example 2
Factorize \(125x^3 - 27y^3\)
\begin{equation*} \begin{split} & 125x^3 - 27y^3 \\\\ & (5x)^3 - (3y)^3\\\\ & (5x -3 y)(25x^2 + 15xy + 9y^2) \end{split} \end{equation*}
Example 3
Expand \((x + 4)(x^2 - 4x + 16)\)
\begin{equation*} \begin{split} & (x + 4)(x^2 - 4x + 16) \\\\ & x^3 + 4^3\\\\ & x^3 + 64 \end{split} \end{equation*}
Example 4
Expand \((m - 3)(m^2 + 3m + 9)\)
\begin{equation*} \begin{split} (m - 3)(m^2 + 3m + 9) &= (m - 3)(m^2 + 3m + 3^2)\\\\ (m - 3)(m^2 + 3m + 9)& = m^3 - 3^3\\\\ (m - 3)(m^2 + 3m + 9)& = m^3 - 27 \end{split} \end{equation*}
Exercise