Number of Fixed points of an odd degree polynomial

Question

Let $p(x)$ be a polynomial of degree $2n+1$ with real coefficients. then $p(x)$ has

(I) exactly $2n+1$ fixed points

(II) at least one fixed point

(III) at most one fixed point

(Iv) $n$ fixed points.

Let $p(x)=a_0+a_1 x+\dots a_{2n+1}x^{2n+1}$ for fixed points we have $p(x)=x$ i.e $a_0+(a_1-1) x+\dots a_{2n+1}x^{2n+1}=0$, so $(I)$ is true?

Answer 1

$p(x)=x^5+x+1$

Make $p(x)=x$ we get $x^5+1=0\Rightarrow x=-1$ (If by fixed point you mean only real)

This tells there is only one fixed point which says :

first option is false as there is only one fixed point... (we should have $5$ for that to be true)

fourth option is false as there is only one fixed point.. (we should have $2$ for that to be true)

Take $p(x)=x^3-x$ consider $p(x)=x$ i.e., $x^3-x=x\Rightarrow x^3-2x=0\Rightarrow x(x^2-2)=0$

which has three fixed points.

So, Only thing you can say is :

Any odd degree polynomial with real coefficient will have at least one fixed point.

Answer 2

HINT: Look at $p(x)=x^3+x+1$. $\quad$