Question

What is Factoring Completely?

Answer 1

For factoring polynomials, "factoring" (or "factoring completely") is always done using some set of numbers as possible coefficient.

We say we are factoring "over" the set.

`x^3 -x^2-5x+5` can be factored
over the integers as `(x-1)(x^2-5)`

`x^2-5` cannot be factored using integer coefficients. (It is irreducible over the integers.)

over the real numbers `x^2-5 = (x-sqrt5)(x+sqrt5)`

One more:
`x^2+1` cannot be factored over the real numbers, but over the complex numbers it factors as
`x^2+1=(x-sqrt(-1))(x+sqr(-1))`

Also written: `(x-i)(x+i)`