To determine if a polynomial has real solution I have the following polynomial : $x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$
I must determine if this polynomial has at least 1 real solution and justify why. We have a theorem which says that all polynomials with real coefficients can be decomposed in a product of polynomials of real coefficients with degree 1 or 2.
So this means we have four scenarios :
Factors : 2+2+2+1 , 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1
In all these cases, we have atleast one factor of degree 1, so there is atleast one real solution in each case. What do you think ?

Since
$$x^7+x^6+x^5+x^4+x^3+x^2+x+1=\frac{x^8-1}{x-1}$$
we can use the fact that $x^8-1$ has roots at $-1$ and $1$ to get the root of $x=-1$. 
