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.

The Intermediate Value Theorem establishes existence:  there is at least one real root.
Notice that $p(0) = -2 < 0$ and $p(1) = 7 > 0$.  Since $p$ is continuous, the I.V.T. guarantees a number $c$ such that $p(c) = 0$.  (In fact, we know that $0 < c < 1$.)
Rolle's Theorem establishes uniqueness:  there is at most one real root.
Why?  Suppose that there were two roots $a, b \in \mathbb{R}$.  Since $p$ is differentiable, Rolle's Theorem guarantees a number $c \in (a, b)$ where $p'(c) = 0$.  What's wrong with that?  The derivative
$$
p'(x) = 5x^4 + 3x^2 + 7 > 0
$$
for all $x \in \mathbb{R}$.  Why?  It's quadratic in $x^2$ and its discriminant $(3)^2 - 4(5)(7) < 0$.
