8.4 - Factoring by grouping

This is probably the hardest factorable form to identify. It is largely a matter of trial and error and about the only thing that can be said for sure is that the expression to be factored has to have at least four terms. The idea is that it may be possible to arrange the terms into groups, each of which can be factored separately. This may cause the groups to have a new common factor which can be factored out. (This is a lot of maybe’s!) Here are some examples:



Example:   Factor  x y + 3 x + 2 y + 6

The first two terms form a group and the last two terms form another group. Factor x out of the first two terms and 2 out of the last two terms:
x(y + 3) + 2(y + 3).
Now the two groups have a common factor of y + 3. Factor out that common factor. This gives:
(x + 2) (y + 3).




Example:   Factor   a 2 c + a 2 d − c − d

The first two terms form a group and the last two terms form another group. Factor a 2 out of the first two terms and a − sign out of the last two terms:
a 2(c + d) − (c + d).
Now the two groups have a common factor of c + d. Factor out that common factor. This gives:
(a 2 − 1)(c + d).



  Algebra Coach Exercises