Actually the quadratic formula is a great example of exactly what I mean. I was taught it as a way to solve for roots of a second degree polynomial when "completing the square" didn't work. The terms in the quadratic formula are defined as the coefficients in the polynomial and the answer is written in the form "x = +-___". Looking back at that chapter in math, that along with the rational roots test for finding zeros in a higher order polynomial were genuine wastes of time. I don't mean the learning the intuition, I mean a question asking to list every single possible root given a second or higher order polynomial.

I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic?

The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution".

Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college.

I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well.

When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel.

I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities.

I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.