### Short Division of Polynomials

Li Zhou
2009 The College Mathematics Journal
When I teach, I always tell my students to ask me "why". So when I was teaching synthetic division in precalculus, my student Heysel Marte promptly asked why the algorithm worked. I told her that it was an excellent question, thought for a moment, and then presented the following sketch to the class. "Suppose that we want to divide ax 3 + bx 2 + cx + d by x − k. Then we are seeking an answer of the form As a casual comment, I pointed out the caution in the textbook about the limitations of the
more » ... lgorithm, but after the class, I asked myself why the idea above could not be extended to divisors of higher degrees. Then I realized that it COULD! Suppose that we want to divide ax 6 + bx 5 + cx 4 + dx 3 + ex 2 + f x + g by x 2 − kx − l. Then we are seeking an answer of the form Expanding the right-hand side and comparing coefficients we see that m = a, n = b + mk, o = c + nk + ml, p = d + ok + nl, q = e + pk + ol, r = f + qk + pl, and s = g + ql. We now display these relations in a format of synthetic division.