A research engineer determines that the stability of a feedback loop in a control system depends on the roots of the cubic polynomial P(y)=8y3+18y2+y−6P(y) = 8y^3 + 18y^2 + y - 6P(y)=8y3+18y2+y−6.
Use the factor theorem to prove that (y+2)(y + 2)(y+2) is a factor of P(y)P(y)P(y).
Hence, using algebraic methods, express P(y)P(y)P(y) as a product of three linear factors.
A specific operating mode of the loop requires the condition P(tanϕ)=0P(\tan \phi) = 0P(tanϕ)=0. For the range π2<ϕ<π\frac{\pi}{2} < \phi < \pi2π<ϕ<π, determine all possible values of ϕ\phiϕ, giving your answers to three significant figures.
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.