Question 362440
"Im lost!

Solve the problem.

Find k such that f(x) = x4 + kx3 + 2 has the factor x + 1."



am sure this problem is f(x) = x^4 + kx^3 + 2, and we need to find k so that
x + 1 is a factor



divide:
          x^3 + (k - 1)x^2 - x + 1
x + 1 --> x^4 + kx^3 + 2
          x^4 + x^3
               (k - 1)x^3       + 2
               (k - 1)x^3 + x^2    
                          - x^2     + 2
                          - x^2 - x
                                  x + 2
                                  x + 1
                                      1


x^3 + (k - 1)x^2 - x + 1 + 1/(x + 1) was result of division
(x^3 + (k - 1)x^2 - x + 1 + 1/(x + 1))(x + 1)
x^3(x + 1) + (k - 1)(x + 1)x^2 - x(x + 1) + x + 1 + 1
x^4 + x^3 + (k - 1)(x^3 + x^2) - x^2 - x + x + 2
x^4 + x^3 + kx^3 + kx^2 - x^3 - x^2 - x^2 + 2
x^4 + kx^3 + kx^2 - 2x^2 + 2
x^4 + kx^3 + (k - 2)x^2 + 2
set k - 2 = 0
k = 2
to get x^4 + 2x^3 + 2


check:
           x^3 + x^2 - x + 1
x + 1  --> x^4 + 2x^3 + 2
           x^4 + x^3
                 x^3  + 2
                 x^3  + x^2
                      2 - x^2
                     -x^2 - x
                      2 + x
                      x + 1
                          1