Am I right in thinking $\dfrac{x^{2}}{ax+b}$ is an improper rational expression? If so, can someone help me figure out how to write it as the sum of a polynomial and proper rational expression?
I have not a clue.
Am I right in thinking $\dfrac{x^{2}}{ax+b}$ is an improper rational expression? If so, can someone help me figure out how to write it as the sum of a polynomial and proper rational expression?
I have not a clue.
I'll do the first few steps; here's hoping you'll catch on to what I'm doing:
$\begin{align*} \frac{x^2}{ax+b}&=\frac{ax^2}{a(ax+b)}\\ &=\frac{ax^2}{a(ax+b)}+\frac{bx}{a(ax+b)}-\frac{bx}{a(ax+b)}\\ &=\frac{(ax+b)x}{a(ax+b)}-\frac{bx}{a(ax+b)}\\ &=\frac{x}{a}-\frac{b}{a}\frac{x}{ax+b}\\ &=\frac{x}{a}-\frac{b}{a}\left(\frac{ax}{a(ax+b)}+\frac{b}{a(ax+b)}-\frac{b}{a(ax+b)}\right)\\ \end{align*}$
Can you take it from here?
Hint: Use polynomial long division to divide $x^2$ by $ax+b$ and you will get a result of the form $(ax+b)(P)+R$. Now divide through by $ax+b$ and you have the desired form.