Question

Let $P_n(x) = 1 + 2x + 3x^2$ + ..... + $(n + 1)x^n$ be a polynomial such that $n$ is even. Then the number of real roots of $P_n(x) = 0$ is

• A. $0$
• B. $n$
• C. $1$
• D. none of these

Answer 1

Answer: (A)

$0$

Answer 2

If $n$ is even, then there is no change of sign in this expression $\therefore$ there is no negative real root of $f(x)$ Hence there is no real root When $x > 0, P_n(x) > 0$ and $\therefore \, P_n(x) = 0$ can have no $+ve$ real root.