Let $p$ be a polynomial with real coefficients such that $\int_0^1 p(t) d t=0$. Which of the following statements is always true?

• A

  $p$ has no roots in the interval $[0,1]$.
• B

  All roots of $p$ lie in the interval $[0,1]$.
• C

  $p$ has exactly one root in the interval $[0,1]$.
• Correct answer
  D

  $p$ has a root in the interval $[0,1]$.

Hence, the correct answer is option: 'D'.