Let's think of a division of natural numbers. Share 7 out of 5 means get the quotient 1 and the rest 2. We can write:
Now let's think about the division of the polynomial A (x) by the nonzero polynomial B (x)which generates the quotient Q (x) and the rest R (x).
In this division:
- A (x) it is the dividend;
- B (x) is the divisor;
- Q (x) is the quotient;
- R (x) is the rest of the division.
The degree of R (x) must be less than the degree of B (x) or R (x) = 0.
When A (x) is divisible by B (x), we say the division is exact, that is, R (x) = 0.
Example 1
Determine the quotient of 
:
Resolution
- We divide the highest degree dividend term by the highest divisor term. The result will be a quotient term:
- We multiply x² per B (x) and subtract the product from A (x), getting the first partial rest:
- We divide the highest degree term of the first partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:
- We multiply -2x per B (x) and subtract the product from the first partial remainder, obtaining the second partial remainder:
- We divide the highest degree term of the second partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:
- We multiply 1 per B (x) and subtract the product from the second partial remainder:
Since the degree of rest is less than the degree of divisor, the division is closed.
We verified that:
Example 2
Determine the quotient of 
:
Resolution
We easily verify that:
In these two examples, we use the key method to effect the division of polynomials.
By the examples we find that:
quotient degree = dividend degree - divider degree
Next: Rest of the Theorem and D'Alembert's Theorem
