If k - x is a factor of the expression x^2 + n k^2, what is the value of n? (assume k ≠ 0)

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Multiple Choice

If k - x is a factor of the expression x^2 + n k^2, what is the value of n? (assume k ≠ 0)

Explanation:
Having a factor means the polynomial equals zero when x takes the value that makes the factor zero. If k − x is a factor, then evaluating at x = k should give zero: P(k) = k^2 + n k^2 = (1 + n)k^2 = 0. Since k ≠ 0, k^2 ≠ 0, so 1 + n must be 0, yielding n = -1. Only this value makes the expression vanish at x = k; other choices leave (1 + n)k^2 nonzero.

Having a factor means the polynomial equals zero when x takes the value that makes the factor zero. If k − x is a factor, then evaluating at x = k should give zero: P(k) = k^2 + n k^2 = (1 + n)k^2 = 0. Since k ≠ 0, k^2 ≠ 0, so 1 + n must be 0, yielding n = -1. Only this value makes the expression vanish at x = k; other choices leave (1 + n)k^2 nonzero.

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