Recently I am studying about theorems regarding to polynomial equations and encounter the lower and upper bounds theorem. Which states that if a<0 and P(a) not equals 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The proof about this statement is provided but I have troubles in understanding it(I do understand about the proof of upper bound one...), I hope someone here can explain and proof about the theorem... Thanks for the help!!!