Bigger Splits

When the numerator has a minus sign, the sign stays with its term when you split the fraction. So $\frac{8x^{3}-4x^{2}}{2x}$ splits into $\frac{8x^{3}}{2x}-\frac{4x^{2}}{2x}$.

Work out each piece. In the first, $\frac{8}{2} = 4$ and $\frac{x^{3}}{x} = x^{2}$, which gives $4x^{2}$. In the second, $\frac{4}{2} = 2$ and $\frac{x^{2}}{x} = x$, which gives $2x$. The quotient is $4x^{2}-2x$.

Divide each term of the numerator by the denominator. Here both coefficients are even and each term contains a factor of $x$, so dividing by $2x$ leaves a polynomial. The original fraction is defined only for nonzero $x$.