I’ve perused the notes and they heavily rely on problem solving to learn the material. It seems like a Moore method style of exposition which I really like. Arnold is a master of mathematical writing and teaching.
We no longer teach the formulas for solving cubic and quartic polynomial equations. Nowadays we just teach linear and quadratic equations. For the most part every algebraic equation you can solve by hand that we teach in basic algebra is an equation that can be reduced to a linear equation or a quadratic equation. It’s amazing how many applied problems can be accomplished by this reduction. With the rise of computers and easy numerical computations this isn’t so important but imagine a scenario in which the useful applications all involved degree 5 or higher polynomials. Would we have progressed much?
You can always approximate a polynomial around a point with a lower-degree polynomial. They will only diverge farther out. As a result whenever physics produces a high-degree polynomial we can inspect certain behaviors in a lower degree. This doesn't help with every case, but it does enough for the situation that many questions become answerable.
If all you changed was physics, the Taylor series thing would put quadratics right back into their position of importance.
You must be posting from a phone.