Splitting the numerator

Question

Can someone explain how we can get the second fraction by splitting the numerator?

$$\frac{x^3}{x^2+x+1}=x-1+\frac{1}{x^2+x+1}$$

I can get the LHS from the RHS but not the other way around. What are the missing steps?

Answer 1

Hint :

$x^3=(x^3-1)+1$ and $x^3-1=(x-1)(x^2+x+1)$

Answer 2

$\dfrac{x^3}{x^2+x+1}=\dfrac{x^3-1+1}{x^2+x+1}=\dfrac{(x-1)(x^2+x+1)+1}{x^2+x+1}=x-1+\dfrac{1}{x^2+x+1}$

Answer 3

Don't they teach long division in school anymore? sheesh

The remainder becomes the numerator in the fractional part of the mixed fraction $$x - 1 +\frac{1}{x^2+x+1}$$