The operating efficiency of a specialized geothermal heat pump is modeled by the function E(T)=T3+(k−3)T2+10T+cE(T) = T^3 + (k - 3)T^2 + 10T + cE(T)=T3+(k−3)T2+10T+c, where T T\,T is the temperature difference across the system in degrees Celsius, and k k\,k and c c\,c are constants with k>0k > 0k>0.
Given that (T−1)(T - 1)(T−1) is a factor of E(T)E(T)E(T):
Show that k+c=−8k + c = -8k+c=−8.
Given also that when E(T)E(T)E(T) is divided by (T+k)(T + k)(T+k), the remainder is -42:
Show that 3k2+10k−c−42=03k^2 + 10k - c - 42 = 03k2+10k−c−42=0.
Hence find the value of k k\,k and the value of ccc.
Find a quadratic expression g(T)g(T)g(T) such that E(T)=(T−1)g(T)E(T) = (T - 1)g(T)E(T)=(T−1)g(T).