Polynomial long division

In algebra, polynomial long division is an algorithm similar to long division for dividing a polynomial into another polynomial of a larger degree. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.

For any polynomials f(x) and g(x), g(x) being of lesser degree than f(x), there exist unique polynomials q(x) and r(x) such that

Synthetic division will find the quotient q(x) and remainder r(x) given a dividend f(x) and divisor g(x). The problem is written down like this:

When the problem is written, all the terms with exponents less than the largest one must be written, even if their coefficients are zero.

Example

Find:

The problem is written like this (note that the x term is included):

1. Divide the first term of the dividend by the first term of the divisor. Place the result above the bar (x³ ÷ x = x²).

2. Multiply the divisor by the term you just wrote. Write the result under the first two terms of the dividend (x² * (x-3) = x³ - 3x²).

3. Subtract the second term of the result you just got from the second term of the dividend and write the result under both of them. This can be tricky at times, because of the sign. (-12x² - (-3x²) = -12x² + 3x² = -9x²) Then, "pull down" the next term from the dividend.

4. Repeat the last three steps, except this time use the two terms that you have just written as the dividend.

5. Repeat step 4. This time, there is nothing to pull down.

The polynomial above the bar is the quotient, and the number left over (-123) is the remainder.

Synthetic Division

Synthetic division is a method of performing polynomial long division without having to maintain long records of the process of long division as above -- though the processes are still the same. It however, only deals with division by monic linear polynomials.

Performing the same example as before:

We concern ourselves basically with the coefficients. We write

Drop the first coefficient after the bar.

Multiply the dropped number by the number before the bar, and place it in the next column.

Perform an addition in that column.

Repeat the previous two steps, the following is obtained

All the numbers on the last row besides the farthest right correspond to coefficients in the quotient; the last number indicates a remainder. The terms are written with increasing degree from right to left, starting to the left of the remainder with degree 0.

The following corresponds to the result of the division:

x² - 9x - 27 - 123/(x-3)