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).
181 exam-style questions on OCR (MEI) A Level Maths 1.3 Functions, covering 1.3.1 Operations on polynomials, 1.3.2 Factor theorem, 1.3.3 Definition of a function (A-level only), 1.3.4 Composite functions (A-level only), 1.3.5 Inverse functions and their graphs (A-level only), 1.3.6 The modulus function (A-level only), 1.3.7 Inequalities with a modulus sign (A-level only), and 1.3.8 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.