I'm currently studying polynomial factorization and would like to know if there is a general strategy to factor expressions by identifying if they come from a known algebraic identity (such as notable products).
For instance, given an expression like $a^4 + b^2$, it is not immediately obvious whether it can be factored using any known identity (such as sum/difference of powers, squares, cubes, etc.).
I am aware of basic identities like:
- $(a + b)^2 = a^2 + 2ab + b^2$
- $(a - b)^2 = a^2 - 2ab + b^2$
- $(a + b)(a - b) = a^2 - b^2$
- $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
- $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
- $a^4 + 4b^4 = (a^4 + 4a^2b^2 + 4b^4) - 4a^2b^2 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)$
However, when faced with a more general expression (e.g., $x^4 - 2x^2 + x$ or $a^4 + b^2$), I find it hard to know whether it matches any pattern or can be rewritten in terms of a product of simpler polynomials.
My question is: Is there a systematic way or algorithmic method to test whether a polynomial expression can be factored using known identities or notable products? Or is this mostly a matter of experience and pattern recognition?
Any guidance, rules of thumb, or resources for learning how to approach such factorization problems would be greatly appreciated.
