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 ?
 A: Your equation is guaranteed to have at least one real solution because it is a polynomial of odd degree. We can prove this by the Fundamental Theorem of Algebra or by the Intermediate Value Theorem.
FTOA- All polynomials of degree $n$ have $n$ roots, real or complex, and complex roots come in pairs, therefore a polynomial of odd $n$ must have at least one real root, because that one root will be the "odd one out" so to speak. 
IVT- for all polynomials of odd $n$, as $x \rightarrow \infty, \ \ x^n \rightarrow \infty$ and as $x \rightarrow -\infty, \ \ x^n \rightarrow -\infty$. And, since all polynomials are continuous, any polynomial of odd degree must cross the $x$-axis at least once, somewhere. This guarantees at least one root exists.
A: 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 $x=-1$. 
A: Astroman.
Let $ f(x) = x^7+x^6+x^5+x^4+x^3+x^2+x+1 $. Being a polinomial function, $f$ is continuous and differentiable along it's domain, which is $ \mathbb{R} $.
Now, let's take $ f(-2) $ and $ f(0) $:
$ \begin{cases} f(-2) = 85 \\ f(0) = 1  \end{cases} $, or, in another words: $ f(-2) < 0 < f(0) $
Thus, according to the Intermediate Value Theorem, because $ f $ is continuous on $[-2,0] $ and $ f $ is differentiable on $ (-2,0 ) $, it exists a $ c \in (-2,0) $ such that $ f(c) = 0 $, which proves that $ f $ has at least one real root.
Kind regards,
Pedro.
A: $\begin{align*}
x^7+x^6+x^5+x^4+x^3+x^2+x+1 &= (x^4+1)(x^3+x^2+x+1) \\
    &= (x^4+1)(x^2+1)(x+1)
\end{align*}$
