Factorising: Take Out the Common Factor

A rectangular garden bed has area 6x + 9 — so what could its length and width be? Turning an area back into "length × width" means splitting it into factors, and the quickest route is to spot what every part has in common and pull it out to the front. That reverse move is factorising.

Expanding a bracket multiplies what is outside into each term inside. Factorising is that same move run in reverse: instead of pulling a bracket apart, you look at a sum of terms, spot what they all have in common, and pull that shared factor back out to the front.

So expanding turns 3(2x + 3) into 6x + 9; factorising turns 6x + 9 back into 3(2x + 3). It looks small, but this reverse move is a master key: it unlocks simplifying expressions, solving equations, and a huge amount of the algebra still ahead of you.

6x + 9 = 3(2x + 3)

The method

To factorise a sum of terms, work through three steps:

Find the highest common factor of the numbers — the biggest whole number that divides every coefficient.
Find the highest common power of any shared letter — if every term contains at least an x, an x comes out too.
Write that common factor outside a bracket, and inside write what is left of each term after you divide it out. Then check by expanding back.

For 6x + 9, both 6x and 9 are divisible by 3, and only the first term has a letter, so the common factor is 3:

6x + 9 = 3 \times 2x + 3 \times 3 = 3(2x + 3)

This is just the distributive law read backwards. Multiply the bracket out again to check: 3 \times 2x = 6x and 3 \times 3 = 9, giving 6x + 9 — back where we started. Get into the habit of that check every time; it catches nearly all mistakes.

Worked example — a common letter

A shared factor need not be a number. In x^2 + 5x, both terms contain an x (remember x^2 = x \times x), so x comes out:

x^2 + 5x = x \times x + x \times 5 = x(x + 5)

Check by expanding: x \times x = x^2 and x \times 5 = 5x, giving x^2 + 5x. Correct.

Worked example — numbers and letters together

Factorise 12ab - 8a. Split it into the number part and the letter part. The numbers 12 and 8 share a highest common factor of 4; both terms contain an a (but only one has a b), so an a comes out too. The full common factor is 4a:

12ab - 8a = 4a \times 3b - 4a \times 2 = 4a(3b - 2)

Check: 4a \times 3b = 12ab and 4a \times 2 = 8a, so expanding gives 12ab - 8a. Back to the start — the factorising is right.

See it as an area

Picture 6x + 9 as the area of a rectangle split into two pieces — a 6x part and a 9 part. Both pieces are the same height, and that shared height is the common factor. Read the height off the side, and the two widths along the bottom spell out the bracket. Step through it.

See it explained

Khan Academy factors a sum by pulling out the greatest common factor — the distributive law in reverse:

Two errors sink most attempts at factorising — guard against both:

Take out the HIGHEST common factor, not just any factor. Writing 12x + 8 = 2(6x + 4) is only partly factorised — 2 is common, but so is 4, and the bracket (6x + 4) still has a common factor hiding inside it. The finished answer is 4(3x + 2). If the bracket you leave behind can still be factorised, you did not take enough out.
Do not lose a term — a "1" is not nothing. Every term inside the bracket, expanded back, must rebuild the original. When a whole term is the common factor, it leaves a 1 behind, not a blank. So 3x + 3 = 3(x + 1), not 3(x) — because 3 \times 1 = 3, and dropping the 1 would throw the +3 away.

Expanding and factorising are the two directions of one skill — and being able to travel both ways is what makes algebra genuinely powerful. Factorising is the essential first step for solving quadratics (once an expression is a product of brackets, you set each factor to zero), and it is how you simplify algebraic fractions — you factorise top and bottom, then cancel the common bracket.

Here is the wild part: the whole security of RSA cryptography — the maths that protects online banking and messaging — rests on the fact that factorising a truly enormous number back into its prime pieces is fiendishly hard, even for a supercomputer. Multiplying is easy; un-multiplying is hard. Spotting the hidden common structure in something, the very thing you are practising here, turns out to be one of the deepest skills in mathematics.