2. Given that $\alpha$ is a solution to the equation $x^{3}+3 x-1=0$. Then the line $y=\alpha x+1-\alpha$ definitely does not pass through ( ).
(A) the first quadrant
(B) the second quadrant
(C) the third quadrant
(D) the fourth quadrant
2. D.

When $x \leqslant 0$, $x^{3}+3 x-10$. From $x^{3}+3 x-1=0$, we get $\alpha^{2}=\frac{1}{\alpha}-3$.
Therefore, $\frac{1}{\alpha}-3>0$, which means $\alpha<\frac{1}{3}$. Thus, $0<\alpha<\frac{1}{3}$.
Hence, the line $y=\alpha x+1-\alpha$ definitely does not pass through the fourth quadrant.