characteristic equation of transfer function

Question

$$\frac{K}{s(s+1)(s+5)}$$

Find the characteristic equation of this transfer function. The book gives this answer:

$$\frac{K}{s(s+1)(s+5)} +1=0$$

or

$$s^3 +6s^2 +5s +K =0.$$

I don't get how the book gets this equation. Where did the 1 come from? And how did it simplify into the second equation?

Answer 1

$\frac{K}{s(s+1)(s+5)}+1=0$

FOIL bottom:

$\frac{K}{s(s^2+6s+5)}+1=0\rightarrow\frac{K}{s^3+6s^2+5s}+1=0$

Move 1 to the right:

$\frac{K}{s^3+6s^2+5s}=-1$

Move denominator to the right:

$K=-1*(s^3+6s^2+5s)=-(s^3+6s^2+5s)$

Divide the negative to the K:

$s^3+6s^2+5s=-K$

Move the K:

$s^3+6s^2+5s+K=0$

Cheers! -Shahar