<s>[INST] 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?
 [/INST] Another way is as follows: the function $$f(x)= x^4 + x^3 \sin(x) + x^2 \cos(x)$$ is an even function (since the cosinus is even and the sinus is odd), hence it is enough to see at $x\ge 0$. The derivative  $$f’(x)=(4+\cos(x))x^3+2x^2\sin(x) +2x\cos(x)$$ is clearly positive for $x$ positive. Consequently $f(x)$ is increasing for $x\gt 0$. Thus the only root is the obvious $x=0$.
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