# Thread: Find the the number of multiple roots of a equation

1. ## Find the the number of multiple roots of a equation

Let $f(x)= 1+ x + \frac{x^2}{2!} + \frac{x^3}{3!} +...+ \frac{x^{2n}}{(2n)!}.$
How many multiple roots does the equation $f(x)=0$ have?

Can someone pls explain how to solve this? Your help is very much appreciated.

2. ## Re: Find the the number of multiple roots of a equation

Originally Posted by shiny718
Let $f(x)= 1+ x + \frac{x^2}{2!} + \frac{x^3}{3!} +...+ \frac{x^{2n}}{(2n)!}.$
How many multiple roots does the equation $f(x)=0$ have?
Suppose $a$ is a multiple root of $f(x)=0$, then $f(a)=0$ and $f'(a)=0$. This implies $f(a)-f'(a)=a^{2n}/(2n)!=0$ or equivalently $a=0$. But $f(0)\neq 0$, that is, $f(x)=0$ has no multiple roots.