<s>[INST] How many distinct real root does the equation $x^{4}-x^{3}\cdot\sin(x)-x^{2}\cdot\cos(x)=0 $ have? How many distinct real root does the equation $x^{4}-x^{3} \cdot \sin(x)-x^{2} \cdot \cos(x)=0 $ have? 
Is there any quick solution(less than 2 minutes)?
 [/INST] You can factor out $x^2$ to get $x^2(x^2-x \sin x - \cos x ) = 0$. So, clearly, there is a root of multiplicity $2$ at $x=0$. Letting $f(x) = x^2-x \sin x - \cos x $, we note that $f$ is even, $f(0) <0$, $\lim_{x\to\infty} f(x) = \infty$ and $f'(x)> 0$ when $x>0$. Consequently $f$ has one real root in $(0,\infty)$.
So the answer is 3 distinct real roots.
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