I really can’t think of a good reason why synthetic division is still in the Algebra 2/PreCalculus curriculum. When I was in high school (at my HS, I was in the last class not to use graphing calculators), synthetic division was a handy shortcut because we had to do tons (often enough for a whole page for one problem) of divisions to test potential zeros for higher-order polynomials. However, with technology as it is, I can think of better ways to spend the 2-3 days it would take to cover synthetic division and finding rational zeros by hand.
I do still teach polynomial division, because it is useful for finding polynomial asymptotes of “improper” rational functions. But for the synthetic division shortcut, unless I’m doing tons and tons of polynomial division problems (and only the case where we’re dividing by linear factors at that), I really can’t think of a good reason to take the time to do it. Plus, I have a really bad memory and always have trouble remembering the proper sign for the number on the outside of the synthetic division box (r when the divisor is x-r…).
At my school, synthetic division is taught in Algebra 2, so I’ve been able to use that reason to skip it in PreCalculus. However, our local community college lists it as a topic that is taught and covered on the final in PreCalculus. So, if I have students who opt to do the college in the high school program, I do need to cover it in PreCalculus…
Please let me know if there’s something I’m missing, or why you do or don’t do synthetic division.