Disprove by counterexample: If $f(0),f(1),f(2)$ are in $\mathbb{Z}$, then $a,b,c,d$ are in $\mathbb{Q}$

Question

Let $f(x) = ax^3 + bx^2 + cx + d.$

Is it sufficient to disprove the implication by showing that at least one variable is not in $\mathbb{Q}$? Or do we need to show that $a,b,c,d$ are all not in $\mathbb{Q}$?

Here's my rationale for showing at least one is not in $\mathbb{Q}$. If $f(0$) is in $\mathbb{Z}$, then $d$ is in $\mathbb{Z}$ because $f(0) = a(0) + b(0) + c(0) + d$.

Answer 1

Consider $a = \sqrt{2}$, $b = -3\sqrt{2}$, $c = 2\sqrt{2}$, and $d=0$. Then $f(1) = \sqrt{2} -3\sqrt{2}+2\sqrt{2} = 0$ and $f(2) = 8\sqrt{2}-12\sqrt{2}+4\sqrt{2} = 0$. So $a,b,c \notin \Bbb Q$ but $f(0), f(1), f(2)\in \Bbb Z$.

Answer 2

From $f(0)\in\Bbb Z$ we have $d\in \Bbb Z$ and from $f(1)\in\Bbb Z$ we get $A=a+b+c\in\Bbb Z$ and finally from $f(2)\in\Bbb Z$ we get $B=8a+4b+2c\in\Bbb Z$. Hence for example if we choose $A=B=0$ we get $3a+b=0$ so $c=2a$ and we can choose for a counterexample

$$a=\pi, b=-3\pi, c=2\pi, d=m\in \Bbb Z$$