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Let $V=\{f(x) \in \mathbb{R}[x] \mid f(0)=0\},$ viewed as a real vector space. Consider the following assertions:

$\text{(I)}$ $V$ contains three linearly independent polynomials of degree $2$ .
$\text{(II)}$ $V$ contains two linearly independent polynomials of degree $3$.

Which of the following statements is correct?

  1. Both $\text{(I)}$ and $\text{(II)}$ are true.
  2. $\text{(I)}$ is true, but $\text{(II)}$ is false.
  3. $\text{(I)}$ is false, but $\text{(II)}$ is true.
  4. Neither $\text{(I)}$ nor $\text{(II)}$ is true.
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