This is a good general tool for less-mathematically-deep folks to keep in their pocket: look for well-behaved objects that do something nice under the operation you're interested in. Typical "well-behaved" objects do things like stay where they are, or end up as a constant multiple of themselves, or end up as 0 or 1, or something like that. Then try to represent everything else in terms of those objects, so that you can take advantage of their convenient behavior. Less-difficult examples include:

- Prime factorization: primes have nice properties, and you can turn every integer into a product of primes (polynomial factorization is an extension of this idea) and work with the nice prime properties

- Vector spaces: basis vectors have nice properties, so you write vectors as sums of them and do operations on the coefficients instead of the vectors themselves

- The exponential function: it's the unique function with f'(x) = f(x), so you try to turn everything else into exponentials anytime you have to solve some painful differential equation because you know those terms will go away

- Fixed points in dynamical systems: if you don't want to analyze how arbitrary things change, find the points that don't, then think of the other points as (fixed point) + (small perturbation) and reduce your work to handling the perturbation

- Taylor series: polynomials are easy, smooth functions are hard, so turn your smooth function into a polynomial and do polynomial things with it