Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options. (1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points (2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots (3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root (4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order (5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order
Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options.\\
(1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points\\
(2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots\\
(3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root\\
(4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order\\
(5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order