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Notes -
Polynomial division is IMHO a curiosity.
Long division crops up every time you need want to split the bill.
Polynomial division might crop up in the wild if you want partial fraction decomposition, which I guess you might want if you are dealing with rational functions and want to numerically evaluate them or calculate their anti-derivatives. While I am sure that rational functions have their uses, my gut feeling is that they are both too narrow to pop up in physics a lot (where you will frequently have square roots placed so that your functions can not easily be transformed into rational functions) and too inconvenient to be preferred for empirical models.
Factorization of an integer is a hard but finite problem. Factorization of a polynomial is in general just not possible exactly. You can test if 7, 13, 17, 19, 23, ... etc happen to divide 3071 to factorize it. You can not test if x-1, x-1.1, x-sqrt(42+sqrt(42)) etc are factors of 5x5+4x4+7x**2-2x-2, because there are countably many algebraic numbers which could be a root.
I think that we learned both polynomial division and solving quadratic equations around grade eight. Solving quadratics in something which I would call bloody useful. Quadratic functions are the first non-trivial functions students can tackle, and quadratic equations pop up all the time in high school physics.
I strongly disagree with the sentiment that math skills which are less readily automated are more valuable. To grok (I'm reclaiming that word) how multiplication and division work doing long multiplication and division is definitely more useful than just using a calculator. Nobody needs the numeracy to be actually excellent at these operations any more. Anyone whose job actually requires them to multiply five-digit numbers will hopefully have the good sense not to try that by hand.
My more general point might be that I do not want students to be excellent at applying any algorithm. They will always suck very hard compared to the simplest of computers. Still, it is useful to demonstrate that you can apply an algorithm, even if it is just at toy-sized problems.
Also, applying a pre-learned algorithm is not math. Some algorithms (e.g. solving an equation for a variable) are genuinely useful in proofs, and thus are valid technical skills to learn to be able to engage in math, same as being able to write symbols with a pencil. And of course, 'can you apply an algorithm halfway reliably?' is also a good way to check if someone has a basic understanding of the algorithm in question (even if it does not probe if they understand why it works), which is why rotating trees by pencil is a staple in CS exams.
Still, for school math, I feel that 'can apply pre-learned algorithms' should earn a passing grade, not an actually good grade, which should require thinking.
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