Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).
Answer Details
To factorize the given expression, we can use synthetic division or long division to divide \(f(x)\) by \(x - 3\) which is one of its factors as given. The result of this division is a quadratic expression that can be factored easily.
Using synthetic division, we get:
\begin{array}{c|cccc} & 2 & -3 & -11 & 6 \\ \hline 3 & & 6 & 9 & -6 \\ & & & -6 & 27 \\ \hline & 2 & 3 & -2 & 21 \end{array}
Thus, \(f(x) = (x - 3)(2x^{2} + 3x - 2)\). Now we need to factorize the quadratic expression \(2x^{2} + 3x - 2\).
We can use the quadratic formula or factorization by grouping to factorize the quadratic expression. Factorization by grouping is simpler and can be done as follows:
\begin{align*}
2x^{2} + 3x - 2 &= 2x^{2} + 4x - x - 2 \\
&= 2x(x + 2) - 1(x + 2) \\
&= (2x - 1)(x + 2)
\end{align*}
Therefore, the complete factorization of \(f(x)\) is:
$$f(x) = (x - 3)(2x - 1)(x + 2)$$
Hence, the correct option is (C) (x - 3)(x + 2)(2x - 1).