In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, polynomial long division is an algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 for dividing a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 by another polynomial of the same or lower degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

, a generalised version of the familiar arithmetic technique called long division

Long division

In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.

Example

Find

The problem is written like this:


The quotient and remainder can then be determined as follows:

1. Divide the first term of the numerator by the highest term of the denominator (meaning the one with the highest power of x, which in this case is x). Place the result above the bar (x³ ÷ x = x²).



2. Multiply the denominator by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the numerator (x² · (x − 3) = x³ − 3x²).



3. Subtract the product just obtained from the appropriate terms of the original numerator (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath ((x³ − 12x²) − (x³ − 3x²) = −12x² + 3x² = −9x²) Then, "bring down" the next term from the numerator.
   



4. Repeat the previous three steps, except this time use the two terms that have just been written as the numerator.



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.


The long division

Long division

In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

 algorithm for arithmetic can be viewed as a special case of the above algorithm, in which the variable x is replaced by the specific number 10.

Division transformation

Polynomial division allows for a polynomial to be written in a divisor–quotient form which is often advantageous. Consider polynomials P(x), D(x) where degree(D) < degree(P). Then, for some quotient polynomial Q(x) and remainder polynomial R(x) with degree(R) < degree(D),
This rearrangement is known as the division transformation, and derives from the arithmetical identity .

Factoring polynomials

Sometimes one or more roots of a polynomial are known, perhaps having been found using the rational root theorem. If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x - r)(Q(x)) where Q(x) is a polynomial of degree n–1. Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.

Likewise, if more than one root is known, a linear factor (x – r) in one of them (r) can be divided out to obtain Q(x), and then a linear term in another root, s, can be divided out of Q(x), etc. Alternatively, they can all be divided out at once: for example the linear factors x– r and x – s can be multiplied together to obtain the quadratic factor x² – (r + s)x + rs, which can then be divided into the original polynomial Q(x) to obtain a quotient of degree n – 2.

In this way, sometimes all the roots of a polynomial of degree greater than four can be obtained, even though that is not always possible. For example, if the rational root theorem can be used to obtain a single (rational) root of a quintic polynomial, it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a quartic polynomial

Quartic function

In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...

 can then be used to find the other four roots of the quintic.

Finding tangents to polynomials

Polynomial long division can be used to find the equation of the line that is tangent to a polynomial at a particular point. If R(x) is the remainder when P(x) is divided by (x – r )² — that is, by x² – 2rx + r ² — then the equation of the tangent line to P(x) at x = r is y = R(x) (regardless of whether or not r is a root of the polynomial).

See also

• Polynomial remainder theorem
  Polynomial remainder theorem
  In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial f\, divided by a linear divisor x-a\, is equal to f \,.- Example :...
  
  
• Synthetic division
  Synthetic division
  Synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the formx - a,\ but the method generalizes to division by any monic polynomial...
  
  , a more concise method of performing polynomial long division
• Ruffini's rule
  Ruffini's rule
  In mathematics, Ruffini's rule allows the rapid division of any polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1809. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor. The Horner scheme is a fast algorithm for dividing...
  
  
• Euclidean domain
  Euclidean domain
  In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
  
  
• Gröbner basis
  Gröbner basis
  In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
  
  
• Greatest common divisor of two polynomials
  Greatest common divisor of two polynomials
  Informally, the greatest common divisor of two polynomials p and q is the largest polynomial that divides both p and q evenly. The definition is modeled on the concept of the greatest common divisor of two integers, the greatest integer that divides both...