Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.
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Tags: algebra, polynomial, number theory, trinomial
Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.