$$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0$$ is a polynomial equation. $$P(0)$$ is a root of the equation $$P(x) = 0$$. Then

Solution

$$P(0) = a_0$$, and we are told this value is a root, so $$P(a_0) = 0$$, i.e. $$P(P(0)) = 0$$ and $$a_0$$ is a root. Nothing forces $$P(0) = 0$$; for example $$P(x) = x^2 + x - 2$$ has $$P(0) = -2$$ and $$P(-2) = 4 - 2 - 2 = 0$$, so $$P(0)$$ is a root while $$P(0) \ne 0$$. Hence options b, c and d are correct.

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