SOLUTION: Im lost! Solve the problem. Find k such that f(x) = x4 + kx3 + 2 has the factor x + 1.

Question 362440: Im lost!
Solve the problem.
Find k such that f(x) = x4 + kx3 + 2 has the factor x + 1.
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
"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

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