SOLUTION: Find all zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) P(x) = x^4 − x^2 + 2x + 2

I will factor the polynomial, using the grouping method


    P(x)  =  x^4 − x^2 + 2x + 2 = (x^4 − x^2) + (2x + 2) = x^2(x^2-1) + 2(x+1) = x^2*(x+1)*(x-1) + 2(x+1) =

               // now I see that there is common factor (x+1), and I factor it out

        = (x+1) * (x^2*(x-1) + 2) = (x+1) * (x^3 - x^2 + 2).


Thus, I just extracted one root x= -1 and separated one relating factor (x+1).


Now I will work with the polynomial of the degree 3 in parentheses.


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

               // now I see that there ios common factor (x+1), and I factor it out

         = (x+1)*(x^2 - 2(x-1)) = (x-1)*(x^2 - 2x + 2).


So, I separated one more root x= -1 and the relating factor (x+1).


Now I will work with the quadratic polynomial in parentheses.

It has the roots

     =  =  = .


ANSWER.  The roots are -1, -1,  1+i  and  1-i.