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). It is a generalisation of standard long division. Polynomial long division is particularly useful when trying to integrate fractions using partial fractions.
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.