Find the number of real roots of the polynomial $f(x)=x^5+x^3-2x+1$    ( I think similar questions have already been asked. However, I am not sure whether the exact question has already been raised by some user. I urge the members of this mathematical community to mark my question as duplicate if it has already been asked. I tried an attempt below:-)    What I attempted:-  I know a little bit of theory of equations but not all. While I was in the examination, I did not ponder much more on any theorems relating to real roots. Rather I tried to do it with some intuition. We have $f(1)=1$,$f(0)=1$,$f(2)=1$, $f(\frac{1}{2})=\frac{5}{32}$,$f(-\frac{1}{2})=\frac{59}{32}$,$f(-2)=-35$ It just helped me to trace a root between $-1$ and $-2$. Then I tried to study the nature of the graph.    Here $f'(x)=5x^4+3x^4-2$ Let $f'(x)=5x^4+3x^2-2=0$     $\Rightarrow 5y^2+3y-2=0$  (where $x^2=y$) $\Rightarrow (5y-2)(y+1)=0$ Therefore, $y=\frac{2}{5}$,   or $y=-1$ (It is not of our interest, as we are dealing with real roots)   Therefore $x=\sqrt{\frac{2}{5}},-\sqrt{\frac{2}{5}}$ These two values divides the real line into three disjoint subsets $(-\infty,-\sqrt{\frac{2}{5}})$, $(-\sqrt{\frac{2}{5}},\sqrt{\frac{2}{5}})$, $(\sqrt{\frac{2}{5}},\infty)$    It s quite easy to notice that $f$ is decreasing in $(-\sqrt{\frac{2}{5}},\sqrt{\frac{2}{5}})$ and increasing in $(-\infty,-\sqrt{\frac{2}{5}})$, $(\sqrt{\frac{2}{5}},\infty)$     Again, $f(-\sqrt{\frac{2}{5}})=1.911$ and $f(\sqrt{\frac{2}{5}})=0.089$    Looking at all these details it is clear that $f(x)$ comes close to zero in $(-\sqrt{\frac{2}{5}},\sqrt{\frac{2}{5}})$ but is not equal to zero. There is no possibility for $f(x)$ to be zero in $(\sqrt{\frac{2}{5}},\infty)$   as it is increasing here. Again, we have already discovered one root between $-1$ and $-2$. Apart from that there is of course no root as $f$ is again increasing in $(-\infty,-\sqrt{\frac{2}{5}})$. So, te graph should look something like this  Therefore there is only one real root of the polynomial.    Is my approach correct? Is there any way to do the same thing using some more beautiful as well as advanced method ? (I have just made a rough plot of the graph in my answer sheet). I am not sure if the examiner would expect much better method than this one.
Determine number of real roots of $f(x)=x^{5}+x^{3}-2x+1$.   I used Descartes's rule of sign to determine the number of positive and negative roots; there is only 1 negative (hence minimum 1 real root) and two positive roots. These 2 positive roots can be anything- a complex pair, or two real roots- but what about the other 2 roots? Is there any flaw in my understanding of rule of signs? How can I find the exact number of real roots?  Thanks in advance.
My site as of right now is a static website. I am planning on having a 'recent blog posts' section on it, with permalink to the main blog post/blog. example:   http://site.com/blog   I was wondering, because of having a recent blog posts section on the homepage, should I just have the entire website powered by WordPress and just add the homepage (and other pages, such as contact) as a word press 'page', and call recent blog posts on it?  If I do it that way, am I still able to have the blog in /blog?  I was also thinking of just adding the whole WordPress files to the /blog directory and call recent blog posts from the static file...  I am new to Wordpress, so I am a bit confessed here.  What is the best option? Which words in a title should be capitalized? I am indian citizen on H1b visa in US. I am planning to visit France in Oct 2014. Can I apply for Schengen visa now about 3/4 months in advance. Are there any restrictions that on early you can apply for Schengen visa. 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