x^3 + 4x^2 - 3x -12 divided by x^2 - 3?

Answer 1

Essentially, you want to eliminate the highest power from the numerator and continue to do so until you can't do it anymore.

So, for the first step, we want to eliminate the x^3 term. The way we do that is subtract x * (x^2 - 3). So, we get:

x^3 + 4x^2 - 3x - 12 - (x^3 - 3x)
= 4x^2 - 12

Now, we want to eliminate the 4x^2 term. We do that by subtracting 4 * (x^2 - 3). So, we get:

4x^2 - 12 - (4x^2 - 12) = 0

So, since we did subtracted x * (x^2 - 3) and 4 (x^2 - 3) and got 0, we know that x^3 + 4x^2 - 3x - 12 = x * (x^2 - 3) + 4 (x^2 - 3).

Factor out the (x^2 - 3) to get:

x^3 + 4x^2 - 3x - 12 = (x + 4) * (x^2 - 3)

Answer 2

x^3 + 4x^2 - 3x -12 = (x^3 - 3x) + (4x^2 - 12) =
x*(x^2 - 3) +4*(x^2 - 3) = (x + 4)*(x^2 - 3) . Now it is obvious that
((x + 4)*(x^2 - 3)/(x^2 - 3) = x + 4.