Usually, after a few days of practicing long division, the teacher will introduce a faster way of completing polynomial division problems, called

*synthetic*division. Essentially, this method allows the student to write less by keeping track of the variables in columns. Often students are resistant to this new method, simply because it is new. However, after practice, they usually concede that it is easier.

Unfortunately, synthetic division only works when dividing by a monic, linear polynomial such as $x-3$. Once we start dividing by quadratic polynomials such as $x^2-3x+5$ most textbooks do not provide any such shortcut...

Today, I offered my students two options. Students could practice dividing by quadratic polynomials using long division, or, if they felt confident in that skill, they could opt to learn a synthetic division technique for quadratic polynomials that was not in the book. Four students had strong enough math skill and had sufficient interest to take me up on the offer to see if the synthetic shortcut for linear polynomials could be extended to quadratics.

Below is the same problem worked out using both long division (left) and synthetic division (right). If you're familiar with synthetic division from high school, you might try to figure out how the procedure works.