Question:
Find the value of $k$ if $x-3$ is a factor of $k^{2} x^{3}-k x^{2}+3 k x-k$
Solution:
Let $f(x)=k^{2} x^{3}-k x^{2}+3 k x-k$
From factor theorem if x - 3 is the factor of f(x) then f(3) = 0
⟹ x - 3 = 0
⟹ x = 3
Substitute the value of x in f(x)
$f(3)=k^{2}(3)^{3}-k(3)^{2}+3 k(3)-k$
$=27 k^{2}-9 k+9 k-k$
$=27 k^{2}-k$
= k(27k - 1)
Equate f(3) to zero, to find k
⟹ f(3) = 0
⟹ k(27k - 1) = 0
⟹ k = 0 and 27k - 1 = 0
⟹ k = 0 and 27k = 1
⟹ k = 0 and k = 1/27
When k = 0 and 1/27, (x - 3) will be the factor of f(x)