# What is Factoring Completely?

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.

${x}^{3} - {x}^{2} - 5 x + 5$ can be factored
over the integers as $\left(x - 1\right) \left({x}^{2} - 5\right)$

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

over the real numbers ${x}^{2} - 5 = \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right)$

One more:
${x}^{2} + 1$ cannot be factored over the real numbers, but over the complex numbers it factors as
${x}^{2} + 1 = \left(x - \sqrt{- 1}\right) \left(x + s q r \left(- 1\right)\right)$

Also written: $\left(x - i\right) \left(x + i\right)$