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Question
Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.
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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