What is Factoring Completely?

1 Answer
Apr 18, 2015

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)#