How can the techniques of factoring be extended to work with higher-degree polynomials?

Explanation:

Any techniques that you're familiar with can be applied to polynomials of any degree. These might include ...

use of the rational root theoremuse of Descartes' rule of signsuse of any algorithms you're aware of for finding bounds on rootsgraphingfactoring by groupinguse of "special forms" (for example, difference of squares, sum and difference of cubes, square of binomials, expansion of n-th powers of binomials)guess and checkmaking use of turning points

Each root you find can be factored out to reduce the degree of the remaining polynomial factor(s).