ISI MMA-2015

Question

If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and  $n\;(\geq1)$ respectively, satisfy

                                                      $f(x^{2}+1) = f(x)g(x)$ $,$

for every $x\in \mathbb{R}$ , then

(A) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) \neq 0$

(B) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) = 0$

(C) $f$ has $m$ distinct real roots

(D) $f$ has no real root.

Comments

I think option D

---

I also think (D) should be answer but not sure about it.

---

Solution given in the answer section

Answer 1

Option D

Comments

you have written "if they are equal then $x_{0}$ has to be imaginary". Can you please explain this line. Complex numbers can't be compared. right ?

---

x0 will be imaginary because