If $(x - 1)$ is a factor of the polynomial $3 x^{3} - 2 x^{2} + k x - 6$ , then what is the value of $k$ ?

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Solution

Since $(x - 2)$ is a factor of the polynomial $p (x) = 3 x^{3} - 2 x^{2} + k x - 6 = 0$ , by factor theorem, we have $p (2) = 0$
$\Rightarrow 3 (2)^{3} - 2 (2)^{2} + k (2) - 6 = 0$
$3 (8) - 2 (4) + 2 (k) - 6 = 0$
$24 - 8 + 2 k - 6 = 0$
$2 k = - 10$
$k = - 5$ .

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