Factoring

Factoring means writing a sum as a product. That sounds roundabout, but it is often the key: a product can be cancelled down and simplified, and it is usually easier to keep working with than a long sum. It is the reverse path of multiplying out, and with a little practice you spot the right split at once.

The common factor

If every term holds the same number or variable, you can pull it in front of the bracket. In $6x + 9$ both parts share the 3: $6x + 9 = 3 \cdot (2x + 3)$. You can check this any time by multiplying out again. If the original sum comes back, the split was correct.

Finding the greatest factor

The most useful one is the greatest common factor, because it makes the bracket as simple as possible. Both 12 and 8 can be divided by 2 and by 4, so the greatest common factor is 4. If you pull out only the 2, a common factor is still sitting inside the bracket. In $4a + 4b + 4c$ the 4 sits in every term: $4a + 4b + 4c = 4 \cdot (a + b + c)$.

Binomial formulas in reverse

Some terms have no common factor and can still be factored, namely with the binomial formulas. A difference of squares is especially rewarding: $a^2 - b^2 = (a + b)(a - b)$. So $x^2 - 9$ becomes the product $(x + 3)(x - 3)$, because 9 is the square of 3.