<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] The idea provided by @JasonM is correct. You should be able to solve the problem as follows.
Notice that the cubic polynomial is monic, meaning that the coefficient of $x^3$ is $1.$ This is nice because now we can write the polynomial as $(x - a)(x - b)(x - c)$ in factored form. Now, we can equate this to the given form:
$$(x - a)(x - b)(x - c) = x^3 + ax^2 + bx + c$$
$$x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc = x^3 + ax^2 + bx + c.$$
We can see from the equation that $-abc = c,$ meaning that $ab = -1.$ We also have that $a + b + c = a,$ meaning that $b + c = 0,$ or $b = -c.$ With $ab + bc + ca = b,$ substituting values in, we get $-1 - c^2 + 1 = - c,$ which can be rearranged to $c^2 - c = c(c - 1) = 0.$ The solution $c = 0$ cannot be accepted because in that case, $b = c = 0$ and the roots are not distinct. But if $c = 1,$ then $b = -1$ and $a = 1,$ in which case $a = c.$ So there is $\boxed{\text{No solution}}$ to the requested problem.
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