Question

How do you factor completely `P(x)= x^3-2x^2+x-2`?

Answer 1

Factored over the real numbers: `(x-2)(x^2+1)`

Factored over the complex numbers: `(x-2)(x+i)(x-i)`

Explanation:

We can factor by grouping:

`x^3+x-2x^2-2=x(x^2+1)-2(x^2+1)=`

`=(x-2)(x^2+1)`

This is all we can factor over the real numbers, but if we include complex numbers, we can factor the remaining quadratic even further using the difference of squares rule:

`x^2+1=x^2-i^2=(x+i)(x-i)`

This gives the following complex factoring:

`(x-2)(x+i)(x-i)`