1985 Putnam A6

If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real
coefficients $a_i$, then set
$$\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.$$
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with
real coefficients such that

(i) $g(0)=1$, and

(ii) $\Gamma(f(x)^n) = \Gamma(g(x)^n)$

for every integer $n \geq 1$.

如果 $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ 是实数多项式

系数$a_i$，然后设置

$$

\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2。

$$

令 $F(x) = 3x^2+7x+2$。通过证明找到多项式 $g(x)$

实系数使得

(i) $g(0)=1$，并且

(ii) $\Gamma(f(x)^n) = \Gamma(g(x)^n)$

对于每个整数 $n \geq 1$。