SOLUTION: factor the sum of cubes 1³ + x³

Factor: 

1 + x³

There are two ways to do it.

First method: Long division:

Write it in descending powers and put in placeholders 
for the x² and x terms:

x³ + 0x² + 0x + 1

This can be divided evenly by x + 1

            x² -  x + 1
x + 1)x³ + 0x² + 0x + 1
      x³ +  x²
           -x² + 0x
           -x² -  x
                  x + 1
                  x + 1
                      0 

So you see that it factors as divisor times quotient:

    (x + 1)(x² - x + 1) 

Second method:

But most people just learn the rule for factoring
sum and difference of cubes, so they won't have
to do the long division like I did above.

The sign between the cube terms is the same sign
as the sign in the first factor and opposite
the sign in the second factor.

A³ ± B³ = (A ± B)(A² ∓ AB + B²)

1³ + x³ = (1 + x)(1² - 1x + x²)

        = (1 + x)(1 - x + x²)

That's the same as (x + 1)(x² - x + 1).

Edwin