Find the value of $k$ if $x - 3$ is a factor of $k^{2} x^{3} - k x^{2} + 3 k x - k$ .

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Solution

$x - 3 = 0 \Rightarrow x = 3$ is a root
$K^{2} λ^{3} - k x^{2} + 3 k x - k = 0$
$27 k^{2} - 9 k + 9 k - k = 0$
$27 k^{2} - k = 0$
$k (27 k - 1) = 0$
$k \neq 0, k = \frac{1}{27}$

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