Why is it useful to factor out the GCF first when factoring?

What signals you that factoring by grouping is the best method to use when factoring a problem? How would you know if it is factorable or prime?

Step 1: Determine the greatest common factor of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.

Step 2: Factor out (or divide out) the greatest common factor from each term.

Step 1: Determine the greatest common factor of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.

Step 2: Factor out (or divide out) the greatest common factor from each term.

Factor by grouping: 6x2 + 3x + 20x + 10.

This is a polynomial written with four terms that don't have a single common factor among them. However, the first two terms have a common factor (3x), and the last two terms have a common factor (10). This situation doesn't answer all of our wildest factoring dreams, but we'll take it.

By pulling out the common factors for each pair of terms, we can rewrite the original polynomial like this:

3x (2x + 1) + 10 (2x + 1)

These two terms now have a common factor of (2x + 1). Seems like we should be able to do something with that information, don't you think? In fact, we can pull out this common factor and rewrite the polynomial again:

(3x + 10) (2x + 1)