Answer in Brief

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

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#### Solution

Let `f(x) = k^2 x^3 - kx^2 + 3kx - k` be the given polynomial.

By the factor theorem,

(*x* − 3) is a factor of *f*(*x*) if f (3) = 0

Therefore,

`f(3) = k^2 (3)^3 - k(3)^2 + 3k(3) - k = 0`

\[\Rightarrow 27 k^2 - 9k + 9k - k = 0\]

\[ \Rightarrow 27 k^2 - k = 0\]

\[ \Rightarrow k\left( 27k - 1 \right) = 0\]

\[ \Rightarrow k = 0 \text { or k } = \frac{1}{27}\]

Hence, the value of k is 0 or `1/27`.

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