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

Given:

Given expression is $k^2x^3 - kx^2 + 3kx - k$.

$(x - 3)$ is a factor of $k^2x^3 - kx^2 + 3kx - k$.

To do:

We have to find the value of $k$.

Solution:

We know that,

If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.

Therefore,

$f(3)=0$

$\Rightarrow k^2(3)^3-k(3)^2+3k(3)-k=0$

$\Rightarrow 27k^2-9k+9k-k=0$

$\Rightarrow 27k^2-k=0$

$\Rightarrow k(27k-1)=0$

$\Rightarrow k=0$ or $27k=1$

$\Rightarrow k=0$ or $k=\frac{1}{27}$

The values of $k$ are $0$ and $\frac{1}{27}$.