Given that&nbsp;\( P(x) = x^5 + ax^4 + bx^3 + c^2 + dx + e\) is divisible by both&nbsp;\(x - 2\) and \( (x + 1)^2 \) Also, \( P(0) = 6\), \(P'(-1) = 0\). Find the polynomial. (Very challenging; combines repeated factors with derivative conditions.)

Given that \( P(x) = x^5 + ax^4 + bx^3 + c^2 + dx + e\) is divisible by both \(x - 2\) and \( (x + 1)^2 \)
Also, \( P(0) = 6\), \(P'(-1) = 0\).
Find the polynomial.
(Very challenging; combines repeated factors with derivative conditions.)

Polynomials, Cubic Equations and Partial Fractions

Polynomials, Cubic Equations and Partial Fractions