Number of roots of $x^4 + x^3 \sin(x) + x^2 \cos(x) = 0$ How many roots do the following polynomial have?
$$x^4 + x^3 \sin(x) + x^2 \cos(x)$$
Obviously $0$ is a root. Dividing by $x^2$, I am left with 
$$x^2 + x\sin(x) + \cos(x)$$
How do I know this polynomial has no root?

Look for extremum of $f(x)=x^2 + x\sin(x) + \cos(x)$
$$
\frac{d}{dx}(x^2 + x\sin(x) + \cos(x))=2x+x\cos(x)=x(2+\cos(x))=0
$$
Then the only extremum we have is at $x=0$
We know that $x(2+\cos(x))<0$ for $x<0$ and $x(2+\cos(x))>0$ for $x>0$, also $f(0)=1$, which means that $f(x)\ge1$ for all $x$, hence it have no roots.
