Common Factors and Grouping

Factorising is expanding backwards: you put the brackets back in. For twelve x plus eighteen, find the highest number that divides both terms. Twelve and eighteen share the factors one, two, three and six, and the highest is six. Divide each term by six: twelve x becomes two x and eighteen becomes three. So twelve x plus eighteen is six times the bracket two x plus three. Check by expanding it back.

Letters can be part of the common factor too. Ten p cubed minus fifteen p squared. The numbers ten and fifteen have a highest common factor of five. Both terms contain p, and the lowest power of p is p squared, so that is the most p you can take out. The HCF is five p squared. Divide each term by it: ten p cubed becomes two p, and negative fifteen p squared becomes negative three. The answer is five p squared times the bracket two p minus three.

When the first term is negative, it is usual to take out a negative common factor. Negative six a squared b plus nine a b squared. Six and nine share three, and both terms have at least one a and one b, so the HCF is three a b. Take out negative three a b. Negative six a squared b divided by negative three a b is positive two a. Positive nine a b squared divided by negative three a b is negative three b. The answer is negative three a b times the bracket two a minus three b. Notice that every sign inside has flipped.

When there are four terms and nothing is common to all of them, group them in pairs. Three m plus three n plus k m plus k n. The first pair has a common factor of three: three times m plus n. The second pair has a common factor of k: k times m plus n. Now both parts contain the same bracket, m plus n, so that bracket is a common factor. Take it out, and what is left is three plus k. The answer is m plus n, times three plus k.

Grouping has one trap. In p q minus four p minus five q plus twenty, the first pair gives p times q minus four. For the second pair, the first term is negative five q, so take out negative five. Negative five q divided by negative five is q, and twenty divided by negative five is negative four, giving negative five times q minus four. The brackets match, so the answer is q minus four, times p minus five. If you took out positive five, you would get five times negative q plus four, and the brackets would not match. Quiz and exercises now.