Question 155001
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Factor 
{{{x^3 + 27}}}

Let's write it this way

{{{x^3 + 3^3}}}

That is the sum of two cubes.

Now what if we dropped those cubes and had {{{x+3}}}
Then suppose we divided that into {{{x^3 + 3^3}}} 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

{{{FIRST^3 + SECOND^3}}}

it factors as 

{{{(FIRST+SECOND)(FIRST^2-FIRST*SECOND+SECOND^2)}}} 

So in the case of

{{{x^3 + 27}}}

You write it as

{{{x^3 + 3^3}}}

Then {{{FIRST = x}}} and {{{SECOND=3}}}

so

{{{(FIRST+SECOND)(FIRST^2-FIRST*SECOND+SECOND^2)}}}

becomes

{{{(x+3)(x^2-x*3+3^2)}}}

or

{{{(x+3)(x^2-3x+9)}}}

Then you don't have to use long division.

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

Suppose, instead it were

{{{x^3 - 27}}}

Let's write it this way

{{{x^3 - 3^3}}}

That is the DIFFERENCE of two cubes.

Now what if we dropped those cubes and had {{{x-3}}}
Then suppose we divided that into {{{x^3 - 3^3}}} 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

{{{FIRST^3 - SECOND^3}}}

it factors as 

{{{(FIRST-SECOND)(FIRST^2+FIRST*SECOND+SECOND^2)}}} 

So in the case of

{{{x^3 - 27}}}

You write it as

{{{x^3 - 3^3}}}

Then {{{FIRST = x}}} and {{{SECOND=3}}}

so

{{{(FIRST-SECOND)(FIRST^2+FIRST*SECOND+SECOND^2)}}}

becomes

{{{(x-3)(x^2+x*3+3^2)}}}

or

{{{(x-3)(x^2+3x+9)}}}

Then you don't have to use long division.

---------

In general

{{{FIRST^3 +- SECOND^3}}}

factors as

{{{(FIRST+-SECOND)(FIRST^2+-FIRST*SECOND+SECOND^2)}}}


Edwin</pre>