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Find the value of k, if x+k is the factor of the polynomial x⁴ −k²x ² +3x−6k

User Arkoudinos
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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.

User Herberth Amaral
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