Here are some examples that show how to use factoring by grouping.

 

Example

Factor by grouping x³ + 2 x² – 3 x - 6

 

Notice there is not anything that every term has in common. Group the problem together to work with it. Think about the same problem as (x³ + 2 x²) + (– 3 x – 6). This is still the same problem only written different. Look at the first grouping (x³ + 2 x²), factor out x² from it to get x² (x + 2).

 

x² (x + 2) + (– 3 x – 6)

 

Look at the second grouping (– 3 x – 6). Factor out a –3.

 

x² (x + 2) – 3 (x + 2)

 

Even though it might seem confusing so far, it is about at the point to finish the problem. The two parentheses are exactly alike (x + 2), whenever they are, re-write it.

 

(x² – 3)(x + 2)

 

Answer! Look carefully between the last two steps and see where the numbers came from. The x² came from in front of the first parentheses and the –3 from in front of the second parentheses.

 

 

 

 

 

 

Example

Factor by grouping y³ - 2 y² – 6 y + 12

 

Group the expression.

 

(y³ - 2 y²) + (– 6 y + 12)

 

Factor y² out of the first and –6 out of the second.

 

y² (y – 2) – 6 (y – 2)

Note- If there would have been a 6 factored out instead of –6 out of the second, it would have resulted in 6 (-y + 2). Then the two parentheses would not have been exactly alike. Recognize this and if an error is made, correct it by factoring out a negative sign.

 

Since the parentheses are exactly alike, rewrite the equation by factor by grouping.

 

(y ² - 6) (y – 2)

 

Answer!

 

 

 

 

Sometimes the change in the appearance of certain expressions is required, in order for the parentheses to look exactly alike. Remember, if the parentheses are not exactly alike, this method cannot be used.

 

First understand this,

-1 (y + 3) = (- y + 3)

 

This may, or may not, seem obvious but it is an application of the distribution rule and very important!

 

 

If -1 (y + 3) = (- y + 3) then (-y + 3)= -1(y + 3). (Just reversed the order)

 

Here is an application of this idea.

 

 

 

 

 

Example

 

Factor by grouping x³ - 3 x² – 6 x + 18

 

The first thing to do is to start to factor. In the first two terms, x³ and 3 x², factor out an x². Out of the last two, – 6 x and 18, factor out a 6.

 

x² (x – 3) + 6 (-x + 3)

 

One thing to notice is that the parentheses are not equal but apply the idea of (-x +3) = -1(x - 3) to the second parentheses.

 

x² (x – 3) + 6 (-1)(x - 3)

 

Simplify a little more.

 

x² (x – 3) - 6 (x - 3)

 

Now there are two parentheses that are the same (x – 3). Rewrite it using factor by grouping.

(x ² – 6) (x - 3)

 

Answer!

 

Note- The first parentheses, (x ² – 6), may look like a difference of two squares but since 6 is not a perfect square it is not. Therefore, do not break it down. If it were a difference of two squares, break it down.