(x-3)(x^2+9)
or
x^3 -3x^2 + 9x - 27
First, let's see about factoring x^4 - 81. Cursory examination indicates that it's the difference of two squares and so it initially factors into
(x^2 - 9)(x^2 + 9)
And the (x^2 - 9) term is also the difference of 2 squares so it too factors into:
(x - 3)(x + 3)
So a partial factorization of x^4 - 81 is:
(x - 3)(x + 3)(x^2 + 9)
The (x^2 + 9) term could be factored as well, but that's not needed for this problem, and so I won't do it.
Now we can divide (x-3)(x+3)(x^2+9) by (x+3). The (x+3) terms will cancel and we get as the result
(x-3)(x+3)(x^2+9) / (x+3) = (x-3)(x^2+9)
We can leave the answer as (x-3)(x^2+9), or we can multiply it out, getting:
x^3 -3x^2 + 9x - 27
