<s>[INST] Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$ Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient
and have all real roots . Prove that $ab \leq 0$
First Let the roots of this polynomial (call it P(x)) be $q,r,s,t$
By Vieta's, $a=-(q+r+s+t)$ and $b=-(qrs+rst+qrt+qst)$ So $ab=q^2(rs+st+rt)+r^2(qs+qt+st)+s^2(qr+qt+rt)+t^2(qr+qs+rs)+4qrst$
But how to find contradiction or use inequality to help ? 
Thanks
 [/INST] Let $p$, $q$, $r$ and $s$ be roots of our equation and
$$p+q+r+s=4u,$$
$$pq+pr+ps+qr+qs+rs=6v^2,$$ $$pqr+pqs+prs+qrs=4w^3$$ and $$pqrs=t^4,$$ where $v^2$ and $t^4$ can be negative.
Thus, $p$, $q$, $r$ and $s$ are roots of the equation:
$$x^4-4ux^4+6v^2x^2-4w^3x+t^4=0,$$ which says that by the Rolle's theorem  the equation
$$(x^4-4ux^4+6v^2x^2-4w^3x+t^4)'=0$$ or 
$$x^3-3ux^2+3v^2x-w^3=0$$ has three real roots. 
Now, if $w^3=0$, so we are done because in this case $b=0$.
But for $w^3\neq0$ we see  that the equation
$$w^3x^3-3v^2x^2+3ux-1=0$$ has three real roots, which says that the equation 
$$(w^3x^3-3v^2x^2+3ux-1)'=0$$ or
$$w^3x^2-2v^2x+u=0$$ has two real roots, which says 
$$v^4-uw^3\geq0.$$
But in our case $v^2=0$.
Id est, $$uw^3\leq0$$ or $$ab\leq0$$ and we are done! 
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