Miscellaneous math question, but why must polynomials be ordered by degree in order to properly divide them by a binomial? I notice arrangement is not a requirement of first degree equations. I can accept this prima facie, however I'd still like to know the specifics and proof as to why this is the case.
Assuming you are specifically asking about writing each polynomial in descending order before carrying out
long division of polynomials, rather than just the convention for writing a polynomial outside of a specific context, I'd say it is just because the long division algorithm wouldn't work otherwise (or at least it would be much harder) unless both divisor and dividend are in descending order. So it's more or less necessary for division, but only
preferred when you are merely
writing a polynomial.
But first degree
equations are an entirely different context from division of polynomial
expressions. (I'm not sure whether you are paying attention to the difference between those words.) This makes me think you aren't primarily asking about division, but about writing expressions (or equations?) generally. In that case, the better question might be why we
don't always bother following the descending order convention for linear expressions, while we do so more consistently with higher-degree polynomials. I'd say that is because with only two terms, most things we do with them aren't hard anyway, and any way we write it will be in either ascending or descending order, which is not true for more terms. Most things we do with more than two terms (factoring, solving equations, ...) are easier in order.
