Why do people think long division of polynomials is complicated ?
I heard this expressed recently and it seems like an odd sentiment. For me, synthetic division is complicated and totally adhoc because it looks nothing like the division of integers I remember from my childhood. In contrast, long division of polynomials is just the same process I know and love from the arithmetic of my youth. To me, long division of polynomials is just a natural extension of long division of numbers.
I generally refuse to cover synthetic division because it's not significantly faster and it fails to cover division by nonlinear polynomials. Why teach something which is weird and not general. Just to make that particular calculation faster ?
In addition, long division of polynomials requires multiplication of polynomials in the calculation. This is a skill which needs reinforcement. In contrast, synthetic division is a one-way street of shifting numbers around without any particular attachment to the larger discussion of polynomial algebra.
What am I missing ?