Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root.
Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies f'(x)=0$ has the root $\alpha$ with multiplicity $r-1,$ I proved that the equation can't have a multiple root of order $\ge3.$
Please help me to solve for $n=2$
$$f(x)=1+x+\dots +\frac{x^n}{n!}$$ so $$f'(x)=f(x)-\frac{x^n}{n!}$$ Suppose $r$ is a multiple root. As you said, then $f'(r)=0$. But $$f'(r)=f(r)-\frac{r^n}{n!}=-\frac{r^n}{n!}=0$$ So we must have $$r=0$$ which is impossible as $$f(0)=1\ne 0$$