In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, computing, linguistics and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

For any polynomials f(x) and g(x), where the degree of f(x) is greater than or equal to the degree of g(x), there exist unique polynomials q(x) and r(x) such that

with r(x) having smaller degree than g(x).

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 a regular (non-algebraic) long division problem:

;

All terms with exponents less than the largest one must be written out explicitly, even if their coefficients are zero. In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...

Example

Find:

The problem is written like this (note that, as explained above, the x term is included explicitly, regardless of the coefficient):

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

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

3. Subtract the product you just obtained from the appropriate terms of the original dividend, and write the result underneath. This can be tricky at times, because of the sign. ((x³-12x²) - (x³-3x²) = -12x² + 3x² = -9x²) Then, "pull down" the next term from the dividend.

4. Repeat the previous 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.

This method is entirely reminiscent of the long division learned in elementary arithmetic classes. In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

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, that is, binomials of the form (x-b), where b is any rational number. In mathematics, Ruffinis rule allows the rapid division of any polynomial by a binomial of the form x − r. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

The change of sign of b when dividing by (x+b) (i.e. -b is written to the left of the bar rather than b) allows us to do additions rather than subtractions that are found in regular long division. This reduces the chance for error when the division is done by hand.

Synthetic division is also called division through Ruffini's rule and was described by Paolo Ruffini in 1809. It has been suggested that this article or section be merged with Root-finding algorithm. ... Paolo Ruffini (Valentano, 1765 – Modena, 1822) was an Italian mathematician and philosopher. ... 1809 was a common year starting on Sunday (see link for calendar). ...

Performing the same example as before:

We concern ourselves basically with the coefficients. We write

Note the change of sign from -3 to 3. 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 result of our division is:

Higher degree synthetic division

The technique of synthetic division described above works only with linear divisors. A similar shortcut method exists for dividing by a quadratic or higher degree monic polynomial.

For instance, let's try to perform the following division:

Create the table as with ordinary synthetic division, using the negative coefficients of the divisor following the leading term. We write

Next, underline the leading coefficient of the right-hand side, multiply it by the left-hand coefficients and write the products beneath the next columns on the right.

Now perform an addition.

Repeat the preceding two steps upon the bottom row of the right-hand side.

The underlined numbers correspond to the coefficients of the quotient, the remaining numbers in the bottom row correspond to the coefficients of the remainder. The terms are written with increasing degree from right to left, with the last terms of the quotient and remainder each having degree 0.

The result of our division is:

See also

• Polynomial remainder theorem
• Gröbner basis