Math 2.0

Commutative algebra Definition

Polynomial long division

Builds on Commutative ringPolynomial ringDegree of a polynomialLeading coefficientDivision algorithm for polynomials

Definition

Let RR be a commutative ring and let f,g∈R[x]f, g \in R[x], where gg has degree mm and its leading coefficient uu is a unit of RR. Polynomial long division of ff by gg is the following procedure.

  1. Start with q=0q = 0 and r=fr = f.
  2. While r≠0r \ne 0 and deg⁡r≥m\deg r \ge m, let cc be the leading coefficient of rr and d=deg⁡rd = \deg r, and replace qq by q+u−1cxd−mq + u^{-1} c x^{d - m} and rr by r−u−1cxd−mgr - u^{-1} c x^{d - m} g.
  3. When r=0r = 0 or deg⁡r<m\deg r < m, stop and return the quotient qq and the remainder rr.

Remarks

Each pass of step 2 removes the leading term of rr, so deg⁡r\deg r drops and the procedure stops. The equation f=qg+rf = q g + r holds at every step, so the output is the unique pair of the division algorithm for polynomials. Example: divide f=x3−2x2−4f = x^3 - 2x^2 - 4 by g=x−3g = x - 3 in ℤ[x]\Z[x]. The steps add x2x^2, then xx, then 33 to the quotient, and the remainder is 55. Check: (x−3)(x2+x+3)+5=x3−2x2−9+5=x3−2x2−4(x - 3)(x^2 + x + 3) + 5 = x^3 - 2x^2 - 9 + 5 = x^3 - 2x^2 - 4. Non-example: in ℤ[x]\Z[x] the procedure does not apply to g=2xg = 2x, because the leading coefficient 22 is not a unit, and the first step for f=xf = x would need the coefficient 12\frac{1}{2}. Division by a monic g=x−ag = x - a is often written in a compact table, called synthetic division; it is Horner's method.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementPolynomial long division, definition, no evidence levelPolynomial long divis…definition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Degree of a polynomial definition
  6. Leading coefficient definition
  7. Monic polynomial definition
  8. Division algorithm for polynomials statement
  9. Polynomial long division definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_long_division
About this page
TagFTAK2F0TR3
URIm2://FTAK2F0TR3
Kinddefinition
WritingW0 stub
MSC13F20, 68W30
DefinesPolynomial long division
UsesCommutative ring, Polynomial ring, Degree of a polynomial, Leading coefficient, Division algorithm for polynomials
Changed2 November 2026

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