what is the cubed polynomial x^3+x+2 factored down?

Question

the question is factored down completely

Answer 1

x³+x+2,

x+1 is a factor because x=-1 produces 0 when substituted.

-1 | 1  0 1    2

      1 -1 1 | -2

      1 -1 2 |  0 = x²-x+2=.

The complete (real) factorisation is (x+1)(x²-x+2).

The quadratic factorises into complex factors, found by applying the formula:

x=(1±√(1-8))/2=(1±i√7)/2, x=½+½i√7, ½-½i√7.

Complete factorisation is (x+1)(x-½-½i√7)(x-½+i√7).