I was working through some algebra problems when I stumbled across a few that I couldn't solve.
1. Let be a polynomial such that \(f(0) = 4, f(1) = 5,\) and \(f(2) = 10\). Find the remainder when \(f(x)\) is divided by \(x(x - 1)(x - 2).\)
2. Let \(f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5\).
Without using long division, find the remainder when \(f(x)\) is divided by \(x^2-1\).
Thank you very much for your help!
