How to know whether a polynomial has factors? It's clear that $a^2 - b^2$ has, and that $a^2 + b^2$ doesn't have. But when a polynomial gets longer and longer, do you have any sort of rule, like you have with natural numbers (when they end in $2$ are even, in $5$ are divisible by $5$, and so on).
 A: We can compare polynomials to numbers and study irreducible polynomials in a manner analogous to how we study prime numbers, and this is often done in abstract algebra.
Assuming that you are more precisely asking about how to tell if a quadratic polynomial is reducible over the real numbers $\mathbb{R}$, you simply check the discriminant. Any quadratic $ax^2+bx+c$, is reducible over the reals if and only if $b^2-4ac>0$. 
So if $a$ is variable and $b$ is constant, $0^2-4(1)(-1)>0 \implies a^2-b^2$ is reducible over the $\mathbb{R}$. Similarly, $0^2-4(1)(1)<0 \implies a^2+b^2$ is irreducible over $\mathbb{R}$.
As an aside, you can also check if an integer-valued polynomial is reducible over $\mathbb{Q}$ using Gauss's Lemma: "When a polynomial is integer valued, one may appeal to Gauss's lemma which states that if the coefficients of a non-constant polynomial $f$ are relatively prime and $f$ is irreducible in $\mathbb{Z}[X],$ then $f$ is irreducible in $\mathbb{Q}[X].$" (from an old post)
A: Every polynomial has factors
$a^2+b^2=(a+ib)(a-ib)$
Note $i=\sqrt{-1}$ anyway there aren't much info on whether a polynomial has real roots, Sturm's theorem,anyway odd degree polynomials always have real roots.For rational roots
Some more help
