Math 2.0

Commutative algebra Theorem

Factor theorem

Builds on Remainder theoremRoot of a polynomial

Statement

Let RR be a commutative ring, let f∈R[x]f \in R[x] and let c∈Rc \in R. Then x−cx - c divides ff in R[x]R[x] if and only if cc is a root of ff.

Remarks

The proof uses the remainder theorem: f=(x−c)q+f(c)f = (x - c) q + f(c). So x−cx - c divides ff exactly when f(c)=0f(c) = 0. For example, 11 is a root of x3−1x^3 - 1, and x3−1=(x−1)(x2+x+1)x^3 - 1 = (x - 1)(x^2 + x + 1). The theorem gives the bound on the number of roots of a polynomial over a field.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementRemainder theorem, statement, no evidence levelRemainder theoremstatementFactor theorem, statement, no evidence levelFactor theoremstatement
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. Polynomial ring definition
  4. Degree of a polynomial definition
  5. Polynomial evaluation definition
  6. Leading coefficient definition
  7. Root of a polynomial definition
  8. Monic polynomial definition
  9. Division algorithm for polynomials statement
  10. Remainder theorem statement
  11. Factor theorem statement

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Factor_theorem, lead section
About this page
Tag0K3MWA7DMC
URIm2://0K3MWA7DMC
Kindstatement, theorem
WritingW1 stated
MSC13F20
StatusThis copy of the site has no build data. Run m2 status 0K3MWA7DMC.
UsesRemainder theorem, Root of a polynomial
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