3. If the quadratic trinomial $2 x^{2}-k x+6$ can be factored into the product of two linear factors, and the coefficients of each factor are integers, then the number of integers $k$ that satisfy the condition is ( )
(A) 8
(B) 6
(C) 4
(D) 3
Answer: (B)
Solution: For the quadratic trinomial $2 x^{2}-k x+6$, the coefficient of the quadratic term is 2, and the constant term is 6. According to this, it can be factored into the following 8 forms:
$$
\begin{array}{l}
(2 x+1)(x+6),(2 x-1)(x-6),(2 x+6)(x+1),(2 x-6)(x-1) \\
(2 x+3)(x+2),(2 x-3)(x-2),(2 x+2)(x+3),(2 x-2)(x-3)
\end{array}
$$

Multiplying the two linear factors obtained from the factorization, the coefficients of the linear terms are $13, -13, 8, -8, 7, -7$, so there are 6 integers $k$ that satisfy the condition.

Therefore, the answer is (B).