Find all polynomials with real coefficients such that $\forall a, b, c \in \mathbb{R}$,

$$
P(a+b-2 c)+P(b+c-2 a)+P(c+a-2 b)=3 P(a-b)+3 P(b-c)+3 P(c-a)
$$
Replacing $a, b, c$ with 0 gives $P(0)=0$ (the constant term is therefore 0). Replacing $b, c$ with 0 next gives $P(-2a)=P(a)+3P(-a)$. We have an equality between polynomials, so the degrees are equal as well as the corresponding coefficients. In particular, the leading coefficients of these polynomials are equal. If the leading coefficient of $P$ is $c_{n}$ (with $P$ of degree $n$), these are $c_{n}(-2)^{n}$ on the left and $c_{n}+3c_{n}(-1)^{n}$ on the right. These must be equal, i.e., $(-2)^{n}=1+3(-1)^{n}$, so $n=1$ or $n=2$. The polynomial is therefore of the form $P(X)=aX^{2}+bX$, and we verify that these polynomials work well.

## Roots and Factorization