Algebra

\(a^2 - b^2 = (a + b)(a - b)\)

Example 1

Factorize \(400 - 9b^2\)

\begin{equation*} \begin{split} 400 - 9b^2 & = 20^2 - (3b)^2\\\\ 400 - 9b^2 & = (20 + 3b)(20 - 3b) \end{split} \end{equation*}

Example 2

Factorize \(x^8 - y^8\)

\begin{equation*} \begin{split} x^8 - y^8& = (x^4)^2 - (y^4)^2\\\\ x^8 - y^8& = (x^4 + y^4)(x^4 - y^4)\\\\ x^8 - y^8& = (x^4 + y^4)((x^2)^2 - (y^2)^2)\\\\ x^8 - y^8& = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2)\\\\ x^8 - y^8& = (x^4 + y^4)(x^2 + y^2)(x + y)(x - y) \end{split} \end{equation*}

Example 3

Expand \((-8 + x)(-8 - x)\)

\begin{equation*} \begin{split} (-8 + x)(-8 - x) & = (-8)^2 - x^2\\\\ (-8 + x)(-8 - x) & = 64 - x^2 \end{split} \end{equation*}

Example 4

Expand \((3x^2 - 2y^2)(3x^2 + 2y^2)\)

\begin{equation*} \begin{split} (3x^2 - 2y^2)(3x^2 + 2y^2) & = (3x^2)^2 - (2y^2)^2\\\\ (3x^2 - 2y^2)(3x^2 + 2y^2) & = 9x^4 - 4y^4 \end{split} \end{equation*}