I suck... so im just going to try and fudge this.
Intergrate the original polynomial:
=b_0/1+b_1/2+...+b_n/(n+1)+ C, where C is some constant.
Now if b_0x/1+b_1x^2/2+...+b_nx^n+1/(n+1)=0,Then the intergral of the original polynomial would result in some constant C. Therefore, there exists at least some area either below or above the X axis
Now if b_0/1+b_1/2+...+b_n/(n+1)=0, that would mean that certain coeffiecients would be negative for the expression to be able to equal 0. This is so because if b_0, b_1.....B_n are all>0, the polynomial would be a positive definite without any roots. Therefore this polynomial has at least one real root.
EDIT: NVM, My answer is wrong. Im an idiot
![Frown :( :(](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)