Factor 27x³ + 8y³

We must memorize the rule for factoring the sum 
and difference of two cubes, which is:

A³ ± B³ = (A ± B)(A² ∓ AB + B²)

Notice that whatever the sign is between the terms 
of the original expression is the same as the sign 
in the first parentheses of the factorization

But the middle term in the second parentheses of
the factorization gets the opposite sign.

The last term in the second parentheses of the
factorization is ALWAYS + .

We now notice that 27x³ + 8y³ can be written as

3³x³ + 2³y³ which can be written as

(3x)³ + (2y)³, the sum of two cubes.

So A = 3x and B = 2y, so the signs of

A³ ± B³ = (A ± B)(A² ∓ AB + B²)

become

A³ + B³ = (A + B)(A² - AB + B²)

which upon substituting for A and B, becomes

(3x)³ + (2y)³ = [(3x) + (2y)][(3x)² - (3x)(2y) + (2y)²] =

[3x + 2y][3²x² - 6xy + 2²y²] =

(3x + 2y)(9x² - 6xy + 4y²)

Edwin