If we were doing it the long way, we would need to consider all the factors of 6:
1 & 6;
−1 & −6;
2 & 3;
−2 & −3
and also many factors of −12:
1 & −12;
2 & −6;
3 & −4
and all the negatives of these.
We could spend a long time finding the correct combination of factors if we use the long method.
Factoring by Grouping Method
Using grouping method, first we find 2 things, the:
(a) Product of the outer 2 terms of the trinomial
(b) Inner number of the trinomial
Then, the only "guess and check" we need to do is to look for 2 numbers whose:
So in our 6x2 + x − 12 example, we are looking for 2 terms whose:
(a) Product is 6x2 × −12 = −72x2 (multiply outer numbers)
(b) Sum is x (inner number).
We try some terms and easily get 9x and −8x.
These are correct since:
Now write the original expression replacing x with (9x − 8x), as follows:
6x2 + x − 12 = 6x2 + (9x − 8x) − 12
We now re-group the right-hand side:
6x2 + (9x − 8x) − 12 = (6x2 + 9x) − (8x + 12)
Now factor each of the bracketed terms:
(6x2 + 9x) − (8x + 12) = 3x(2x + 3) − 4(2x + 3)
On the right-hand side, we notice that each term in brackets is the same, so we can combine them as follows:
3x(2x + 3) − 4(2x + 3) = (3x − 4)(2x + 3)
[What just happened?
If we have 3xA − 4A, we can factor A out of each term, and write (3x − 4)A. This is how grouping method works. You always end up with brackets that have the same terms inside, and these can be factored out.]
So our answer is:
6x2 + x − 12 = (3x − 4)(2x + 3)
Always check your answer by multiplying it out!