Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).
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Correct Answer: Option C
Explanation:
Since f(3) = 0, then (x - 3) is a factor of f(x).
Dividing f(x) by (x - 3), we get \(2x^{2} + 3x - 2\).
\(2x^{2} + 3x - 2 = 2x^{2} - x + 4x - 2\)
\(x(2x - 1) + 2(2x - 1) = (x + 2)(2x - 1)\)
Therefore, \(f(x) = (x - 3)(x + 2)(2x -1)\)
Since f(3) = 0, then (x - 3) is a factor of f(x).
Dividing f(x) by (x - 3), we get \(2x^{2} + 3x - 2\).
\(2x^{2} + 3x - 2 = 2x^{2} - x + 4x - 2\)
\(x(2x - 1) + 2(2x - 1) = (x + 2)(2x - 1)\)
Therefore, \(f(x) = (x - 3)(x + 2)(2x -1)\)