# Polynomial long division

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 x3-6x+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 x3/x=x2. As this is x2, it is placed above the x2 terms as follows:



Then the divisor is multiplied by what we just wrote above the division, i.e. multiply x-3 by x2. 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, -x2-6x-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, -x2 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 x2-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.