jmc

Question

There exist constants c_2, c_1, and c_0 such that x^3 + x^2 - 5 = (x - 3)^3 + c_2 (x - 3)^2 + c_1 (x - 3) + c_0.Find c_2^2 + c_1^2 + c_0^2.

Accepted Answer

Let y = x - 3. Then x = y + 3, and align* x^3 + x^2 - 5 &= (y + 3)^3 + (y + 3)^2 - 5 &= y^3 + 10y^2 + 33y + 31. align*Thus, c_2^2 + c_1^2 + c_0^2 = 10^2 + 33^2 + 31^2 = 2150.