Final answer:
Applying the remainder theorem, we find that k must equal 0 because when substituting x = -k into the given polynomial and simplifying, the equation indicates that k must nullify the term to satisfy the factor condition.
Step-by-step explanation:
To find the value of k when x+k is a factor of the polynomial x⁴ − k²x² + 3x − 6k, we can apply the remainder theorem. This theorem states that if x+p is a factor of a polynomial f(x), then f(-p) = 0. In this case, p is -k, so we need to evaluate the polynomial at x = -k.
First, replace x with -k in the polynomial. The equation becomes:
((-k)⁴ − k²(-k)² + 3(-k) − 6k = 0)
Which simplifies to:
((k⁴ − k´ + 3k − 6k = 0)
As the k´ terms cancel each other out, the equation simplifies to:
((k⁴ − k´ +
3k
− 6k = 0)
Simplifying further, we get:
(-3k − 6k = 0)
Which leads to:
(-9k = 0)
Therefore, k = 0.