Math 2.0

Commutative algebra Proof

Proof of the factor theorem

Builds on Commutative ringRemainder theoremRoot of a polynomialEvaluation homomorphismDivisibility in a commutative ring

Proof

Let RR be a commutative ring, let f∈R[x]f \in R[x] and let c∈Rc \in R.

  1. Suppose that x−cx - c divides ff in R[x]R[x]. Then f=(x−c)hf = (x - c) h for some h∈R[x]h \in R[x].
  2. The evaluation homomorphism at cc is a ring homomorphism. So f(c)=(c−c)h(c)=0⋅h(c)=0f(c) = (c - c) h(c) = 0 \cdot h(c) = 0.
  3. Thus cc is a root of ff.
  4. Conversely, suppose that f(c)=0f(c) = 0. By the remainder theorem, there is q∈R[x]q \in R[x] with f=(x−c)q+f(c)f = (x - c) q + f(c).
  5. Since f(c)=0f(c) = 0, this gives f=(x−c)qf = (x - c) q. So x−cx - c divides ff in R[x]R[x].

Idea

The remainder of ff on division by x−cx - c is the number f(c)f(c). So x−cx - c goes into ff exactly when this remainder is zero, that is, when cc is a root.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first. The path takes the proof Proof of the remainder theorem.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionEvaluation homomorphism, definition, no evidence levelEvaluation homomorphi…definitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionProof of the remainder theorem, proof, no evidence levelProof of the remainde…proofDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementRemainder theorem, statement, no evidence levelRemainder theoremstatementProof of the factor theorem, proof, no evidence levelProof of the factor t…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Ring homomorphism definition
  6. Divisibility in a commutative ring definition
  7. Degree of a polynomial definition
  8. Polynomial evaluation definition
  9. Leading coefficient definition
  10. Evaluation homomorphism definition
  11. Root of a polynomial definition
  12. Monic polynomial definition
  13. Proof of the remainder theorem proof
  14. Division algorithm for polynomials statement
  15. Remainder theorem statement
  16. Proof of the factor theorem proof

Where this leads

142 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Polynomial_Factor_Theorem
About this page
TagWM42HJQBGS
URIm2://WM42HJQBGS
Kindproof
WritingW0 stub
MSC13F20
ProvesFactor theorem
UsesCommutative ring, Remainder theorem, Root of a polynomial, Evaluation homomorphism, Divisibility in a commutative ring
Changed4 November 2026

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