# Number of distinct roots between complex roots among three polynomials

I want to prove that any three relatively prime polynomials $$A, B, C \in \mathbb C[X]$$ verifying $$A+B+C=0$$ have at least $$1+\max(\deg A, \deg B, \deg C)$$ distinct roots in total among each other.

I have tried using the gcd to characterize the number of distinct roots, as for a polynomial $$P$$ its number of distinct roots is given by the degree $$P$$ minus that of $$\gcd(P,P')$$. As $$A, B, C$$ are relatively prime and $$A+B+C=0$$ I can show that $$A$$ and $$B$$ are relatively prime and assume without loss of generality that $$\deg A=\deg B=n$$ and $$\deg A+B \leq n$$. So I want to show that $$ABC=-(A^2B+AB^2)$$ has at least $$n+1$$ distinct roots.

I tried considering the rank of the Sylvester matrices $$S_A, S_B$$ and $$S_{A+B}=S_A+S_B$$ of $$(A,A')$$, $$(B,B')$$ and $$(A+B,A'+B')$$ respectively, but this seems a dead end (inequalities for the rank of the sum being too wide, and it does not exploit that $$A,B$$ are relatively prime).

I have also considered reasoning by induction as the property is clearly true for $$n=1$$, but again this does not seem to yield much... Any ideas?