The power output P P\,P in kilowatts of a specialized wind turbine is modeled by the function P(x)=2x3+kx2−25x+cP(x) = 2x^3 + kx^2 - 25x + cP(x)=2x3+kx2−25x+c, where x x\,x is the wind speed in ms−1\text{ms}^{-1}ms−1 and k,c k, c\,k,c are constants.
At a wind speed of 0.5 ms-1, the turbine produces no power. Show that k+4c=49k + 4c = 49k+4c=49.
It is determined that at a wind speed of 2 ms-1, the power output is exactly c−30c - 30c−30. Determine the value of k k\,k and the value of ccc.
Using algebraic division or otherwise, fully factorise the expression for P(x)P(x)P(x).