Expanding Brackets

$(x+2)(x+3)$ looks frightening, but opening it takes just one law. Factorising packs an expression back into brackets; today we walk the other way and expand, putting every term out in the sunlight.

The distributive law: one host, every guest

Whoever stands outside a bracket must shake hands with every term inside. $3(x+4)$ means 3 times $x$ and 3 times 4:

$a(b+c) = ab + ac$

$3(x+4) = 3x + 12$

As an arrow diagram: from the 3, one arrow points at $x$ , another at 4, and each landing spot keeps its share. Miss one arrow and the answer comes up short.

Two brackets: draw a grid

When two brackets multiply, the rule is everything times everything. Lay out $(x+2)(x+3)$ as a grid:

$\times$	$x$	$+3$
$x$	$x^2$	$3x$
$+2$	$2x$	$6$

Add the four cells together:

$(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Some people call this FOIL — First, Outer, Inner, Last. It is the same job as the grid, wearing different words.

Signs are the disaster zone

The numbers rarely bite; the minus sign does. Two rules of engagement:

A minus sign in front of a bracket is a hidden $-1$ multiplying in, so every term flips: $-(x-4) = -x + 4$ ;
When a bracket carries its own minus, the sign travels with the number: $(x-2)(x+5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10$ .

A minus sign flips everyone inside

The minus in $-(x-4)$ does not govern only the first term. It multiplies $x$ and also $-4$ , so the result is $-x + 4$ , not $-x - 4$ . Check how far the minus reaches before expanding.

Is the expansion right? Substitute a number

Expanding is easy to get wrong — and just as easy to check. Pick any $x$ ; both sides must agree. Try $x = 1$ :

$(1+2)(1+3) = 3 \times 4 = 12 \qquad 1^2 + 5 \times 1 + 6 = 12$

Both sides give 12, so the expansion holds. For extra confidence try $x = 2$ too (both sides give 20). This is the same skill practiced in substitution.

InteractiveExpander

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

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The machine does not care what an expression looks like — only what comes out for the $x$ you feed in. $(x+2)(x+3)$ and $x^2+5x+6$ wear different faces, yet the same input produces the same output. Expanding, at heart, is rewriting the same machine.

Check yourself

Quick quiz

1. What is 3(x + 5) expanded?
2. What is −(x − 4) expanded?
3. What is (x + 2)(x + 3) expanded?