Using the Intermediate Value Theorem and Rolle's theorem to determine number of roots 
Use the Intermediate Value Theorem and Rolle's Theorem to show the that the polynomial $$p(x) =  x^{5} + x^{3} + 7x - 2$$ has a unique real root.

Can someone please give some hints on how to do this question. Thanks in advance.

First, $p\,'(x)=5x^4+3x^2+7$; this is a quadratic in the variable $x^2$, so you can find all of its integer roots. Rolle’s theorem says that if $p(a)=0$ and $p(b)=0$, where $a<b$, then there is an $x$ between $a$ and $b$ such that $p\,'(x)=0$. Thus, if $p(x)$ has more than one real root, $p\,'(x)$ has at least one real root. Finally, remember that if $p(a)<0$ and $p(b)>0$, then the intermediate value theorem insures that there is an $x$ between $a$ and $b$ such that $p(x)=0$. That’s everything that you need to use for the problem.
