I am trying to solve this but I think enough information is not given for example in (A): it will only hold true if we do not take $ \ x \ $ as imaginary .
(All four are given as correct in answer) .
So can anyone help me with this question?
Let quadratic equation $ \ p(x) \ = \ 0 \ $ (where $ \ p(x) \ = \ x^2 + bx + c \ ) \ $ and equation $ \ p(p(p(x))) \ = \ 0 \ $ have a common
root, then which of the following statements is/are correct:
(A) if $ \ b,c \ \in \mathbb{ R}, \ $ then $ \ b^2 – 4c \ \ge \ 0 \ . $
(B) if $ \ p(0) \ = \ 1 \ , \ $ then $ \ p(1) \ = \ 0 \ . $
(C) the equations $ \ p(p(p(x))) \ = \ 0 \ $ and $ \ p(p(p(p(p(x))))) \ = \ \ $0 have at least two common roots.
(D) zero is a root of the equation $ \ p(p(p(p(p(p(x)))))) \ = \ 0 \ . $
Best Answer
Suppose $x_0$ is the common root of the equation $p(x)=0$ and $p(p(p(x)))=0$. Then we have $p(p(p(x_0)))=p(p(0))=p(c)=0$. This means that $c$ is a root of $p(x)=0$.
A) If $b,c \in \mathbb{R}$, then the quadratic will have a real root (namely, $c$), hence it will have both roots real, hence the discriminant is positive.
B) This is just if $c=1$. We know $p(c)=0$.
C) You can see that $x_0$ and $c$ are both common roots here.
D) Can you see why this is true now?