A polynomial in x is x3 + 5x2 – kx – 24. Which of the following is a factor of the given polynomial so that the value of k is 2?

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(x + 4) Explanation: By the factor theorem, (x + 4) is a factor exactly when f(–4) = 0. For f(x) = x3 + 5x2 – kx − 24, f(–4) = (–4)3 + 5(–4)2 – k(–4) – 24 = –64 + 80 + 4k – 24 = 4k – 8 Set 4k – 8 = 0 ⇒ 4k = 8 ⇒ k = 2 Thus, (x + 4) is the factor that gives k = 2.

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