Symbolic Division

Symbolic Division is a method of finding the roots and factors of a polynomial given at least one factor.
This is a function, rather than a program, and can be used in graphs, other functions, and programs.
Rather than returning a list of factors as a result, we return an expression.

# Code

``````:synthdiv(colist,ftr)
:Func
:Local count,ansList,xexp,ansExp
:expr("x")→xexp
:0→ansExp
:newList(dim(colist))→ansList
:colist→ansList
:For count,2,dim(ansList)
:colist[count]+ansList[count-1]*ftr→ansList[count]
:EndFor
:For count,1,dim(ansList)
:ansExp+ansList[count]*xexp^(dim(ansList)-count-1)→ansExp
:EndFor
:ansExp-ansList[dim(ansList)]/xexp+ansList[dim(ansList)]/(xexp-ftr)→ansExp
:Return ansExp
:EndFunc```
```

# What's Going On?

First, this is a function, not a program! To make a function, press F1 in the program editor, choose "3:New", and make sure the "Type:" is "Function"
This takes 2 arguments, a list, and a number. We assume that's what we are given, and don't bother to check the type of the arguments.
The first argument, colist, should be a list of all the coefficients in your expression. The second, ftr, is one single factor or root of your expression.

## Variables

We then set up a few variables for our function.
Functions can read global variables, but cannot write or create global variables!
count is used in our For loops.
ansList is a list of all the factors we want to return
xexp is the expression "x". We use it to create the returned expression.
ansExp is the factors in expression form.
To prevent errors when making our expression, we set our answer expression ansExp to zero.
We create the answer list using newList() with the number of elements in our given colist
In symbolic division, you "drop" your first coefficient. We just store the first element of colist as the first element of our answer list ansList

## Dividing

Now we start a For loop. We start from 2 since we already found our first answer above.
The next line multiplies the previous answer by the factor, and adds that to the current coefficient, storing it in the next element of our answer list.

## Prettifying the Answer

We start another For loop.
Then we add our current expression to the answer list element times "x" to the power of the appropriate power. and store that as our answer expression.

## Fixing an Error

After we prettify the answer, we have a mistake. Since the answer is an expression, we can easily subtract the mistake, and add our correction.

## Returning the Answer

All Functions must return something using Return

# Examples

Here are a few examples of use.

synthdiv({5,.46,84,.50,7},1/5)
5x3-45x2+75x-35

synthdiv({1,5,6},1)
12
-- +x+6
x-1