What is the completely factored form of the expression 27y cubed minus1?

Fortunately, one of the three roots of 27y3 - 1 is really obvious, y = 1/3. Therefore we can pull (y - 1/3) out of the expression giving us:

(y - 1/3)(27y2 + 9y +3).

Using the quadratic formula on the second multiplicative expression gives:

[-9 ± √(-243)]/54 = (-9 ± 9i√3)/54.

So the complete, factored form of 27y3 - 1 is:

(y - 1/3)[y - (9 + 9i√3)/54][y - (9 - 9i√3)/54].