I have a question where the characteristic equation (CE) for the longitudinal motion of an aircraft is given as (4th order polynomial):
$$\lambda^4+6.335\lambda^3+20.496\lambda^2+0.8663\lambda+0.5063=0$$ Of which within it are 2 2nd order polynomials (a short period and a phugoid CE), one of which is also given to be:
$$\approx\lambda^2+0.0346\lambda+0.02497=0$$ And so to find the other 2nd order polynomial, which the form is known as:
$$\lambda^2+a\lambda+b=0$$ Hence:
$$\lambda^4+6.335\lambda^3+20.496\lambda^2+0.8663\lambda+0.5063=(\lambda^2+0.0346\lambda+0.02497)(\lambda^2+a\lambda+b)$$
Of which I had managed to get to:
$$\lambda^4+\lambda^3(a+0.0346)+\lambda^2(b+0.0346a+0.02497)+\lambda(0.0346b+0.02497a)+0.02497b=0$$
Through comparison: $$6.335\lambda^3=\lambda^3(a+0.0346)$$ $$20.496\lambda^2=\lambda^2(b+0.0346a+0.02497)$$ $$0.8663\lambda=\lambda(0.0346b+0.02497a)$$ $$0.5063=0.02497b$$
I can find $b$ to be (also given as the answer):
$$b=\frac{0.5063}{0.02497}\approx20.276$$
But how do I find $a$ from the above equation through substitution and comparison when its all over the place with different values? The answer I was given was: $$a\approx6.616$$
I do not know what I am missing here or what assumptions I need to make.
