TITLE: [True/False]The polynomial $x^4+7x^3−13x^2+11x$ has exactly one real root.
QUESTION [6 upvotes]: [True/False]The polynomial $x^4+7x^3−13x^2+11x$ has exactly one real
  root.

I want to solve it without drawing the graph. Here is my idea. Note that $f(1)=1+7-13+11=6>0$ and $f(-1)=1-7-13-11=-30<0$
So we have at least one real root. Now since degree is $4$ we have $4$ roots but rest three can not be complex as they occur in pairs, so we must have another real root.
So the statement is False
Am I right?
Thanks for reading and all the help.

REPLY [5 votes]: Yes, your result is correct (how to improve the actual argumentation is sufficiently discussed within the comments of this answer). It can be done in another by noting that
$$f(x)=x^4+7x^3-12x^2+11x=x\cdot(x^3+7x^2-12x+11)=x\cdot g(x)$$
$f(x)$ has one real root at $x=0$. Now, $g(x)$ is of degree $3$ and every polynomial of degree $3$ has at least one real root by the Intermediate Value Theorem since $\lim\limits_{x\to-\infty}g(x)=-\infty$ and $\lim\limits_{x\to\infty}g(x)=\infty$. 
Thus, $f(x)$ has at least two real roots.