Draw the amplitude and phase diagram through the transfer function
The trick is to not transform the transfer function the way you did into a sum. Just use the original form

$$G(s) = \frac{A(s-z_1)(s-z_2)\ldots(s-z_n)}{(s-p_1)(s-p_2)\ldots(s-p_n)}$$

From this form, you can read off directly the DC gain, the pole frequencies and the zero frequencies. 

That is enough to draw the Bode plot.