Find the value of k if x - 3 is a factor of

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)