Use Rolle's theorem to find the condition for the polynomial equation to have…

MATHEMATICS · Subjective · A Level · Mathematics · Proof · Proof

Problem

Use Rolle's theorem to find the condition for the polynomial equation

f(x)=0

to have repeated real roots. Hence, or otherwise, prove that the equation

x^n+\frac{x^{n-1}}{1!}+\frac{x^{n-2}}{2!}+\cdots+\frac{x}{n!}=0

cannot have repeated roots.