> To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically, I have to use an iterative method like Newton's either way. Doesn’t you

> To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically, I have to use an iterative method like Newton's either way. Doesn’t your programmer brain want things to run as fast as possible? If you have another weapon in your arsenal for solving polynomial equations, you have an extra option for improving performance. As a trivial example, you don’t call your Newton solver to solve a linear equation, as the function call overhead would mean giving up lots of performance. Also, if you solve an equation not because you want to know the roots of the equation but because you want to know whether they’re different, the numerical approach may be much harder than the analytical one. > I would've thought I'd be more interested in whether they're solvable in some other respect (via iteration, via Newton, with quadratic convergence) That’s fine. For that, you read up on the theory behind various iterative methods.