Dividing polynomials starts with dividing monomials, and dividing monomials boils down to reducing fractions, and reducing fractions? Pshaw, we've been doing that for eons. No big whoop. The fractions have variables now, but so what? We've got this.
The most important thing to remember is that when we divide a variable by itself, it equals 1, just like 5 ÷ 5 = 1 or x ÷ x = 1.
Dividing Monomials by Monomials
Let's look at an example. Remember, fractions are just another way to write division.
Instead of writing 
, we can write yyy (which means 
).
Now we can divide, or reduce, the coefficients and the variables. 
and 
, just like 
.
Simplified, this looks like:
Sample Problem
Divide 125x2y by 150xy2.
For simplicity, we can write this as a fraction:
Now let's write the variables the long way.
Then reduce:
Simplified, it look like this:
Sample Problem
Divide 
.
There are no variables with exponents that we need to write out, so we can go straight into reducing:
Simplify that beast.
Dividing Polynomials by Monomials
We may also need to divide polynomials by monomials. To do this, we need to separate the "fractions" into smaller fractions with just one term in each numerator.
We can rewrite this fraction as:
(Remember, when you add fractions together, you combine the numerators and keep the denominator.)
Now, let's write out the variables the long way:
Then reduce:
Simplified, it looks like this:
Look Out: watch your negative signs in a fraction bar. 
is the same as 
, which is also the same as 
, but it's not the same as 
, which would equal 
.