Post #4008965
2026-07-21 00:50 UTC
Replies (1)
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@bemmesr@mathstodon.xyz 2026-07-21 08:57
@SmartmanApps@dotnet.social it makes sense that the textbook should have very dry material and should rely on a teacher to elucidate its contents to their students. Though, many teachers have a pretty uninvolved style of teaching, where they hand out the material without much comment. As you suggested, I had a particular problem in mind when I wrote my post. We are told that when splitting a complex polynomial fraction, we are to split the denominator into its factors, and that for each of these factors we can create a new fraction with a placeholder variable as its numerator and the given factor as its denominator, and we are given to believe that the sum of these new fractions can be equated to the original fraction, so long as we find the right values for the numerators. This is all fairly intuitive, but I have some trouble when considering the scenario where one of the resulting denominators is an irreducible quadratic. We are told that in such a case, the numerator should be a prototype of linear form, with two place holder variables in the form Ax + B. I've managed to get my hands on something of an explanation for why this is necessary, which is that we want to express as many polynomials as we can in the numerator, but any polynomial of order two or higher will make the fraction representable as the sum of a polynomial and a fraction with an order one or fewer numerator. Supposedly this means we don't need to bother with more than order one. What I don't understand is why we can't just use a single placeholder variable? After all, a variable could well come to be defined as being a polynomial in some other variable, so why doesn't this work?