> To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically Typically when you're solving a polynomial equation in an applied contex

> To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically Typically when you're solving a polynomial equation in an applied context (programming, engineering, physics, whatever), it's because you have modeled a situation as a polynomial, and the information you really want is squirreled away as the roots of that polynomial. You don't actually care about the exact answer. You've only got k bits of precision anyway, so Newton's method is fine. But we're not interested in the solution. We don't particularly care that t = approx. 1.016 is a numerical solution.[0] We're not using polynomials to model a situation. In mathematics, we are often studying polynomials as objects in and of themselves, in which case the kind of roots we get tells us something about polynomials work, or we are using polynomials as a lens through which to study something else. In either case it's less about a specific solution, and more about what kind of solution it is and how we got it. Not to mention, specific polynomials are examples. Instead of t^100 + t + 1 = 7, we're usually looking at something more abstract like at^100 + bt + c. [0] And in the rare case we actually care about a specific root of a specific polynomial, and approximate numerical solution is often not good enough.