Algebra
Mathematica TE can work not only with numbers, but also with algebraic formulas.
In[1]:= (9 c x + 7 c y + 7 d x + 8 d y) + (6 c x + 5 c y + 3 d x)
Out[1]= 
Mathematica TE factors the expression as a product of linear terms.
In[2]:= Factor[%]
Out[2]= 
Here is the formula 
.
In[3]:= 9 (2 + x) (x + y) + (x + y)^2
Out[3]= 
This takes the previous expression, represented by %, raises it to the third power, and expands out products and powers. The result is fairly complicated.
In[4]:= Expand[ %^3 ]
Out[4]= 
Factoring this expression yields a much simpler form.
In[5]:= Factor[ % ]
Out[5]= 
This is a sum of rational functions.
In[6]:= 2 x / 3 + 2 / (x - 1) - 1 / (x - 2)
Out[6]= 
You can put them over a common denominator.
In[7]:= Together[%]
Out[7]= 
Going the other way, you break up a rational function into a sum of fractions with simple denominators.
In[8]:= Apart[1 / (8 + 14 x + 7 x^2 + x^3)]
Out[8]= 
Dividing 
by 
gives this quotient and remainder.
In[9]:= PolynomialDivision[3 x^3 - 5 x^2 + 2 x - 5, x + 3, x]
Out[9]= 
