Factor 


Let's write it this way



That is the sum of two cubes.

Now what if we dropped those cubes and had 
Then suppose we divided that into  by 
long division:


            x² - 3x +  9
     ------------------- 
x + 3)x³ + 0x² + 0x + 27
      x² + 3x²
      --------
          -3x² + 0x
          -3x² - 9x
          ---------
                 9x + 27
                 9x + 27  
                 -------
                       0 

That gives a zero remainder. So you now 
know from that, that

x³ + 27

or

x³ + 3³

factors as 

(x + 3)(x² - 3x + 9)

But if you learn the principle then you
wouldn't have to use long division. Sure, if
you forgot the principle you could use
long division every time.  But you should
memorize the principle to save time.

The principle is

When you have the sum of two cubes



it factors as 

 

So in the case of



You write it as



Then  and 

so



becomes



or



Then you don't have to use long division.

-----------------------------------------------

Suppose, instead it were



Let's write it this way



That is the DIFFERENCE of two cubes.

Now what if we dropped those cubes and had 
Then suppose we divided that into  by 
long division:


            x² + 3x +  9
     ------------------- 
x - 3)x³ + 0x² + 0x + 27
      x² - 3x²
      --------
           3x² + 0x
           3x² - 9x
          ---------
                 9x + 27
                 9x + 27  
                 -------
                       0 

That gives a zero remainder. So you now 
know from that, that

x³ - 27

or

x³ - 3³

factors as 

(x - 3)(x² + 3x + 9)

But if you learn the principle then you
wouldn't have to use long division. Sure, if
you forgot the principle you could use
long division every time.  But you should
memorize the principle to save time.

The principle is

When you have the sum of two cubes



it factors as 

 

So in the case of



You write it as



Then  and 

so



becomes



or



Then you don't have to use long division.

---------

In general



factors as




Edwin