i need to use with Rolle's theorem to prove that this Polynomial has no then 2 Square root, Regardless of the values m and n.
i not find any Square root is can be?
someone can show me the correct way to Solve it ?
thanks.
This problem, as stated, does not require you to find any roots, just determine that roots exist. (And not "square roots". Square roots are specifically roots to the equation $\displaystyle x^2= a^2$.) It's not clear to me what you mean by "no then 2 roots". What does the "then" mean? Do you mean that, for all m and n, it has either 0 or 2 roots, but never 1, 3, or 4?
Rolle's theorem says that if f(a)= f(b)= 0 then there exist some number, c, between a and b, such that f'(c)= 0. You want to show that $\displaystyle x^4- 2x^3+ 6x^2+ mx+ n= 0$. If you take that to be f', then $\displaystyle f(x)= \frac{1}{5}x^5- \frac{1}{2}x^4+ 2x^3+ \frac{m}{2}x^2+ mx+ C$ for some number C. You can always choose C to make f= 0 for a given x.