Prime Polynomials

Not every polynomial factors. Try $x^{2}+x+1$. The only pair with product 1 is 1 and 1, and its sum is 2, not 1. The list of pairs is complete, so there is no factoring with integers. A polynomial that does not factor is called prime, like a prime number.

$x^{2}+100$ is prime too. It looks like a difference of squares, but the sign is wrong. It is a sum of squares, and the sum times difference rule needs a minus. There is no pair with product 100 and sum 0.

To decide whether a monic quadratic is prime, list every pair of integers with the right product. If no pair has the right sum, the quadratic cannot be factored over the integers. Leave it as it is and say that it is prime.