Question 51
Let $f(x)$ and $g(y)$ be two monic polynomials with complex coefficients, and of the same degree $n$, such that
$$
f(x) - g(y) = \prod_{j=1}^{n} (a_j x + b_j y + c_j),
$$
where $a_j, b_j, c_j$ are complex numbers, $1 \le j \le n$. Prove that there exist complex numbers $a, b, c$ such that
$$
f(x) = (x+a)^n + c, \quad g(y) = (y+b)^n + c.
$$
$$
f(x) - g(y) = \prod_{j=1}^{n} (a_j x + b_j y + c_j),
$$
where $a_j, b_j, c_j$ are complex numbers, $1 \le j \le n$. Prove that there exist complex numbers $a, b, c$ such that
$$
f(x) = (x+a)^n + c, \quad g(y) = (y+b)^n + c.
$$