What is Expanding, factoring? — Mathematics

Definition 2.1 High School Mathematics · Chapter 2 — Algebra: Equations and Inequalities

Expanding an expression means transforming products into sums; factoring means the reverse, transforming a sum into a product. The basic rule is distributivity:

k(a + b) = ka + kb, \qquad (a + b)(c + d) = ac + ad + bc + bd .

Examples

Example 2.2

Expand $(2x + 3)(x - 5)$ step by step:

\begin{align*} (2x + 3)(x - 5) &= 2x \times x + 2x \times (-5) + 3 \times x + 3 \times (-5) \\ &= 2x^2 - 10x + 3x - 15 \\ &= 2x^2 - 7x - 15 . \end{align*}

Example 2.4

Used forward (expanding) and backward (factoring):

\begin{align*} (3x + 2)^2 &= 9x^2 + 12x + 4, & x^2 - 25 &= (x+5)(x-5), \\ (x - 4)^2 &= x^2 - 8x + 16, & 4x^2 + 4x + 1 &= (2x + 1)^2 . \end{align*}

Mental arithmetic benefits too: $101 \times 99 = (100+1)(100-1) = 10000 - 1 = 9999$ .

Example 2.6

Factor $E = (2x + 1)^2 - (x - 3)^2$ . This is a difference of two squares $a^2 - b^2$ with $a = 2x+1$ and $b = x-3$ :

\begin{align*} E &= \bigl[(2x+1) + (x-3)\bigr] \times \bigl[(2x+1) - (x-3)\bigr] \\ &= (3x - 2)(2x + 1 - x + 3) \\ &= (3x - 2)(x + 4). \end{align*}

Mind the second bracket: subtracting $x - 3$ means subtracting $x$ and adding $3$ .

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