$f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$
I'm trying to understand why do all polynomials with real roots are factorizable. The explanation relies on the fact that all polynomials which are divided by a first degree polynomial (x-b) where b is a root will have
$(x-b)(a_{n-1}x^{n-1}+\cdots+a_0)+R$ R is a real number
I dont understand why this is so. Why is there no possibility of remainders which are not real numbers?
On a separate note, does this mean that polynomials with no real roots are unfactorizable?