A proof of the existence of real root for odd degree real polynomials by induction After thinking over the flaw in my argument of regarding the proof that every odd degree polynomial with real coefficients has at least one real root, I started devising another proof. The brief idea of the proof is as follows. 
Suppose that $f$ be a polynomial with real coefficients where $f(0)\ne0$, let, $$P(n):= \deg{f}=2n+1\implies \exists \alpha_{2n+1},\beta_{2n+1}\in\mathbb{R}\mid (f(x)<0\ \forall x\le\phi_{2n+1})\land(f(x)>0\ \forall x\ge\psi_{2n+1})$$
Base Case: $P(0)$ is trivially true.
Induction Step: So, we assume that for some $k\in\mathbb{N}$ $P(i)$ is true for all $i$ such that $0\le i\le k$. To prove $P(k+1)$ let $f$ be an arbitrary polynomial of degree $2(k+1)+1$, i.e., $2k+3$, say, $$f(x)=a_{2k+3}x^{2k+3}+a_{2k+2}x^{2k+2}+\ldots+a_1x+a_0$$Now let $x^{2i+1}$ the term with non zero coefficient for some $i$ such that $0\le i\le k$ (if $i=k+1$ then the case is trivial). Then we rewrite $f(x)$ as, \begin{align}f(x)&=x^{2i+2}\left(a_{2k+3}x^{2(k-i)+1}+a_{2k+2}x^{2(k-i)}+\ldots+a_{2i+2}\right)+\left(a_{2i+1}x^{2i+1}+\ldots+a_0\right)\\&=g(x)+h(x)\end{align}Now since $\deg g,\deg h\le 2k+1$, by induction hypothesis we conclude that $f(x)<0$ for all $x<\min \left(\phi_{2(k-i)+1},\phi_{2i+1}\right)$.
Similar argument holds for $f(x)>0$.
 A: You are not proving the existence of a root, but of values where the polynomial is positive (negative).
The proof assumes there exists a nonzero odd coefficient below the highest degree term, which is not correct.  For instance $x^5 - x^4 + 1$ does not inductively reduce to anything of degree $3$ or $1$.
Call an odd degree polynomial "sign controlled" if there is a constant $c$ such that the sign of $p(x)$ is the same as the sign of its highest degree monomial whenever $|x|>c$.  Your argument is that sign-controlled polynomials are closed under the following operations:


*

*multiplication by $x^2$ (doing this several times multiplies by $x^{2k}$)

*multiplication by a nonzero constant

*addition of polynomials whose highest-degree coefficients have the same sign.
and you hope to generate any odd degree polynomial by those operations from polynomials of degree $1$, which are sign controlled.
But these operations can only produce polynomials whose nonzero odd degree coefficients are all of the same sign.  It is not possible to produce polynomials such as $x^3 - x$, so this method will not show that they have the right signs for large values of $|x|$. 
