FACTOR BY GROUPING

Factoring a problem means you want to find the original factors the problem came from. One method is called factoring by grouping. You want to arrange the problem so factor out the factors the parts of the group have in common which is to undo the distributive property.

Problem: Factor the following: 3x³ + 2x² – 3x – 2

Solution:

Look closely at the first two terms: 3x³ + 2x^{2 .} They BOTH have x² as a common factor. Factor x² out of the first two terms:

3x³ + 2x²

x²(3x + 2)

Look at the last two terms: – 3x – 2. They BOTH have a negative sign in common. Factor out the – sign from the last two terms:

– 3x – 2 = – (3x + 2) ß did you notice the sign changes INSIDE the parentheses?

So, the problem becomes:

3x³ + 2x² – 3x – 2

x²(3x + 2)  – 1(3x + 2)

Now you see a common binomial that the group has: This common binomial is 3x + 2. Factor out this common binomial that BOTH parts have:

(3x + 2)( x² – 1)

Now notice the factor x² – 1 is a difference of squares problem. Factor x² – 1.

x² – 1 = (x + 1)(x – 1)

The final answer is: (3x + 2)( x + 1)(x – 1).