<s>[INST] Rational root coefficient I saw this question in my exam recently,

If a, b, c are distinct rational roots of $x^3+ax^2+bx+c=0$, find the values of a, b, c.

Can someone give me a hint or answer? I tried factoring it and then expand it all out, but it doesn't seem very useful as they are all "rational". Can someone help me with it?
 [/INST] Apologies for the length of the argument.  I tried to format it to be more comprehensible.

Multiplying out $(x-a)(x-b)(x-c)=x^3-(a+b+c)x^2+(ab+ac+bc)x-abc$, so we have the equations: 
$1)$ $a=-a-b-c$
$2)$ $b=ab+ac+bc$
$3)$ $c=-abc$
Notice that if $a = 0$ or $b = 0$, then by the $(3)$, $c=0$ contradicting the fact that they must be distinct.  Therefore $a,b\neq 0$.  
Now if $c=0$, then by $(1)$ and $(2)$, $2a=-b$ and $b=ab$, which has the solution $a=1, b=-2, c=0$.  The goal now is to show no other solution exists.  
Consider when $c \neq 0$.  Cancelling the $c$'s in $(3)$ gives us $1=-ab \implies a=-\frac{1}{b}$.  Then we can simplify $(1)$ into $b^2+cb-2=0$ 
$\implies b=\frac{-c \pm \sqrt{c^2+8}}{2}$
and $(2)$ simplifies to $(c-1)b^2-b-c=0$ 
$\implies b=\frac{1\pm\sqrt{4c^2-4c+1}}{2(c-1)}=\frac{1\pm(2c-1)}{2(c-1)}$
for $c\neq 1$ 
(if $c=1$, then $(c-1)b^2-b-c=0\implies b=-1 \implies a=1$, contradicting distinct values again).  
Setting the equations for $b$ equal yields 
$\frac{-c \pm \sqrt{c^2+8}}{2}=\frac{1\pm(2c-1)}{2(c-1)}$
or
$\frac{-c \pm \sqrt{c^2+8}}{2}=\frac{1\mp(2c-1)}{2(c-1)}$
Changing these two equations to the one
$c^2+8=\left(\frac{1\pm(2c-1)}{c-1}+c \right)^2$
makes it slightly easier.  
Taking the $-$ gives us $c=-1 \implies b=-1$ contradicting distinct again, and taking the $+$ gives us the more ugly cubic equation 
$c^3-2c^2+4c-2=0$
By the Rational Root Theorem, one need only test the values $\pm 1, \pm 2$, none of which yield a solution to this cubic, so this cubic has no rational solutions.  
Therefore, the only solution was the first one we found; namely $\boxed{a=1, b=-2,c=0}$.  
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