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.
Practise Edexcel A Level Maths Algebraic Methods with exam-style questions for A Level Maths. 261 questions covering 7.1 Algebraic Fractions, 7.2 Dividing Polynomials, 7.3 The Factor Theorem, 7.4 Mathematical Proof, and 7.5 Methods of Proof, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
