What do you know about polynomials?

Calculus Level 4

There are nn polynomials (P1(x).....Pn(x))\left(P_1(x)..... P_n(x)\right) satisfying the following two equations:

8P(x)=(P(x))2P(x)P(0)=4.\begin{aligned} 8 \cdot P(x) &= (P'(x))^2\cdot P''(x) \\ P(0) &= 4. \end{aligned}

Find the value of i=1nPi(1).\displaystyle \sum_{i=1}^n P_i(1).

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