Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function and $P$ be a polynomial of degree 4 with coefficients in $\mathbb{R}$. If $P(f(x))=0$ for all $x \in \mathbb{R}$, then

• A

  $f(x)=0$ for all $x \in \mathbb{R}$.
• Correct answer
  B

  $f$ is a constant function.
• C

  for all continuous functions $g$, there exists $x \in[0,1]$ such that $P(g(x))=0$.
• Correct answer
  D

  $P$ has at most two roots which do not belong to $\mathbb{R}$.

Hence, the correct answer is options: 'B', 'D'.