A mechanical engineer determines that the stability factor, SSS, of a dampening system is modeled by the function S(ω)=6ω3−5ω2−17ω+6S(\omega) = 6\omega^3 - 5\omega^2 - 17\omega + 6S(ω)=6ω3−5ω2−17ω+6, where ω\omegaω is a frequency parameter.
Use the factor theorem to show that (3ω−1)(3\omega - 1)(3ω−1) is a factor of S(ω)S(\omega)S(ω).
Hence, using algebraic methods, express S(ω)S(\omega)S(ω) as the product of three linear factors.
Solve, for 0<ϕ<π0 < \phi < \pi0<ϕ<π, the equation
6tan3ϕ−5tan2ϕ−17tanϕ+6=0 6\tan^3 \phi - 5\tan^2 \phi - 17\tan \phi + 6 = 0 6tan3ϕ−5tan2ϕ−17tanϕ+6=0giving your answers to 3 significant figures.