Dividing by a Variable Term

When the divisor contains $x$, split the fraction in the same way. Then use the quotient rule on each piece: divide the coefficients and subtract the exponents. For $\frac{6x^{2}+9x}{3x}$, the pieces are $\frac{6x^{2}}{3x} = 2x$ and $\frac{9x}{3x} = 3$.

The quotient is $2x+3$. Look at the second term. In $\frac{9x}{3x}$, the $x$ on top and the $x$ below cancel, because $\frac{x}{x} = 1$. Only the number 3 is left. A term loses its $x$ completely when the divisor has the same power of $x$.

There is a second method. Factor the numerator as $3x\left(2x+3\right)$, then cancel the common factor $3x$ from the top and the bottom, for nonzero $x$. Both methods give $2x+3$. The next section explains how to factor polynomial expressions.