Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

Question

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations?

I've tried setting up a system of equations, but this is supposed to be able to be done by hand, and that became too tedious.

Thanks in advance

Answer 1

set $(q-c)(q- (a+b))(q- (a-b))=0$. This should refer to the same polynomial. Compare the coefficient, we get

$c+2a=3a$

$c(a+b)+c(a-b)+a^2-b^2=b^2$

$-c(a+b)(a-b)=c$

we get $a=b=c=0$