Find the value of k for which the expression \(x^3 - 3x^2 - 7x - k\) will be divisible by \((x - 5)\).
15
By the remainder theorem, a polynomial \(f(x)\) is divisible by \((x - 5)\) exactly when \(f(5) = 0\).
Here \(f(x) = x^3 - 3x^2 - 7x - k\), so substitute x = 5.
Compute each term: \(5^3 = 125\), \(3 \times 5^2 = 75\), and \(7 \times 5 = 35\).
So \(f(5) = 125 - 75 - 35 - k = 15 - k\).
Set this equal to zero: \(15 - k = 0\), giving \(k = 15\).
Hence, the required value of k is 15.
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