Find a polynomial p(x) that simultaneously has both the following properties. (i) When p(x) is divided by x^100 the remainder is the constant polynomial 1. (ii) When p(x) is divided by (x-2)^3 the remainder is the constant polynomial 2.
Though I could find the polynomial by taking derivative (works better because the remainders are simple),I wanted to understand how CRT gives the polynomial.
The solution states the following: x^100 and (x − 2)^3 do not share a common factor, you know without any work that a polynomial with given properties must exist. The same Euclidean algorithm (but now with polynomials) gives a systematic way to find it. In the given problem we could use a different trick because the specified remainders here were rather simple (constants).But there is a conceptual way as well by implementing the Chinese remainder theorem.
I am new and poor with Latex.And thank you in advance.