How do symbolic math software work? As the answer to a question of mine I was referred to a website (see here please)
How can WolframAlpha do it like humans?

A symbolic calculator basically is a fancy calculator, pretty much in line with any other calculator.  You can find a whole string of calculators that basically go from your old 4-function right to mathematica and kindred programs.
For example, some of the more recent calculators allow fraction-input, like 5.1.3 for $5\frac 13$.  Some allow complex numbers and matrices to be put in, by moving the cursor around the field, or whatever.  A polynomial is a array of numbers.
As long at the programmer can implement some function, and communicate this to the user, you can do all sorts of fancy things by programming.  Not all mathematical algorithms work here, though.
There is some fancy maths out there, that variously do and do not work in real time.  A recent project i did was to seek out for which prime $p$ exists, if $p \mid F_n$, then so does $p^2$.  Such ought exist, but the program found none under $20\cdot 120^4$.  The trick was to see if $L_{2p}=3 \pmod{p^2}$, where $L_n$ is the nth lucas number.  
It did find the Heron number $103$, where a triangle $n-1, n, n+1$ is a triangle with integer area, then if $103$ divides any of these numbers, so does $103^2$.
