Question

How do you factor completely `y^3 - 27`?

Answer 1

`y^3-27 =(y-3)(y^2+3y+9)`

Explanation:

One of the standard factor formulas is
`color(white)("XXX")`(x^3-a^3) = (x-a)(x^2+ax+a^2)#

In this case, substituting `y` for `x` and `27 = 3^3` for `a^3`
we get the answer above.

Note that in general we need to check the second factor `(y+3y+9)` for possible additional factors.
In this case the discriminant is `< 0` which tells us no additional (Real) factors are available.