If \((x^4 - 2x^3 + 3x^2 - x + k)\) is a multiple of \((x - 3)\), then find the value of k.
-51
-51 is correct.
By the factor theorem, \((x - 3)\) is a factor of \(p(x)\) only when \(p(3) = 0\).
Substituting \(x = 3\): \(p(3) = 3^4 - 2(3^3) + 3(3^2) - 3 + k = 81 - 54 + 27 - 3 + k = 51 + k\).
Setting \(51 + k = 0\) gives \(k = -51\), so the value of k is -51.
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