The bigger picture is that factorization is decomposition. Depending on the domain, that can mean different things. In geometry, factoring a polynomial in x an

The bigger picture is that factorization is decomposition. Depending on the domain, that can mean different things. In geometry, factoring a polynomial in x and y decomposes its curve into a union of subcurves. The curve of xy = 0 is the union of the curves of x = 0 (vertical line through origin) and y = 0 (horizontal line through origin), so the curve of xy = 0 is a pair of crossing lines with a double point at the origin. Similarly, a complete factorization of f(x) decomposes the intersection of y = f(x) and y = 0 into a union of points (the linear factors). But what about irreducible quadratics like x^2 + 1? From a geometrical perspective, they're an algebraic nuisance that goes away with complex numbers. In linear time-invariant systems, a factorization of a transfer function corresponds to a serial decomposition of the system because of the convolution theorem. The most interesting case is when the system is recursive, so that the transfer function is a general rational function (a ratio of polynomials) and you have feedback loops in the block diagram. A parallel decomposition corresponds to a partial fraction expansion, which is based on division and factorization. These decompositions can also be mixed, so you can take a transfer function that is a ratio of sextics, factor them into a pair of cubic ratios (serial decomposition) and then break each of those halves into partial fractions (parallel decomposition). In calculus, the partial fraction expansion lets you integrate all rational functions once you know how to integrate functions of the form x^n for any integer n, positive or negative. I could give more examples, in probability theory (characteristic functions and moment-generating functions), in combinatorics (generating functions), in the theory of linear differential equations (the Laplace transform diagonalizes shift-invariant operators like differentiation, so differential equations become algebraic equations), etc.