Synthetic division question
why does the answer to not have a fraction under 
but have a fraction? 

To answer this question let us first define what a rational and irrational number is. A rational number is basically a number whose decimals are terminating (ie have an end) or repeating (continuously repeats the same pattern. 

An example of this from your question would be 

5.3 - terminating rational number
5.333333333333 - repeating rational number

Irrational numbers are basically numbers whose decimals are non terminating and non repeating. 

An example would be the square root of 29. When plugged in a calculator this would result in 5.3851648071 
This number goes on and on with no set pattern.