Synthetic division is a standard part of the stereotypical "algebra 2" course in the US (~grade 11) and is normally covered including drill problems and examination.
In my experience, cubics and quartics are in the algebra 2 book (I just checked and they are in mine) but in sections with an asterisk. Typically the asterisk sections are not covered because they are considered more difficult diversions and the time, especially for non-superstar students, is better spent solidifying the unasterisked sections. But most teachers will at least say something verbally like" "There are complicated formulas, sort of like the quadratic solution, that can be used to solve general cubics or quartics. We won't cover them because of time, but they are in the book and you should know that methods exist. However, there is no general algebraic solution of 5th or higher order polynomials."
Note that it is standard also to have teaching, drill, and examination on simple (pencil and paper) numerical estimation methods for finding roots to polynomials.
For what it's worth, I think the quadratic really is more fundamental/important in terms of applied problems. Some people say the reason more applied problems use the quadratic is because via circular logic, the students know the equation and this is probably true in some instances. However, we also routinely get people asking questions for fundamental formulas using cubic or higher equations and there really aren't many. Yes, you can construct problems that have enough complication to get to that, but I would not say they are fundamental in the way that the harmonic oscillator is for instance.
For what it's worth, I do think there is a small benefit to covering cubic and quartic methods but would only bother with very high end classes with time to burn. The benefit is from manipulational skill, not from the methods themselves. But being trained, confident in more elaborate manipulations is helpful when you get into physics problems and the ability juggle Taylor series solutions, Bessel functions, Fourier, or change of coordinate systems (without messing up algebraically) is helpful. [Despite what the math and compsci and Maple lovers say, "19th century mathematics is alive and well in physics departments".]