**Polynomial long division**, also called "polynomial division," is a method in algebra for dividing one polynomial by another, either with the same degree or lower. In short, it simplifies the ratio of two polynomials, f(x) and g(x) as:

where h(x) and k(x) are polynomials, each with a degree less than the degree of f(x).^{[1]} It is a generalisation of standard long division. Polynomial long division is particularly useful when trying to integrate fractions using partial fractions.

## Method

Polynomial long division is best seen through and example. Suppose we wish to simplify:

Here x^{3}-6x+2 is the dividend and x+1 is the divisor. The division is denoted as:

Then the first term of the dividend is divided by the first term of the divisor, in this case x^{3}/x=x^{2}. As this is x^{2}, it is placed above the x^{2} terms as follows:

Then the divisor is multiplied by what we just wrote above the division, i.e. multiply x-3 by x^{2}. This is written below the dividend:

We then subtract this from the dividend and carry down the remaining terms in the polynomial:

The process is then repeated on the new polynomial, -x^{2}-6x-2. This continues until the degree of the first term in the dividend cannot be is less than the degree of the first term of the divisor. For the example above, -x^{2} is of higher degree than x so the process continues:

As -5x has the same degree as x, we perform another iteration:

Now that we have reached a value (3) that has a degree less than that of x, we stop. The polynomial above the dividend, in this case x^{2}-x-5, is the quotient or h(x) as denoted above. The bit at the very bottom, 3, is the remainder or k(x) as denoted above. This means we can write:

This is simpler and makes integration much easier.