Math 2.0

statement theorem

A polynomial over a field has at most as many roots as its degree

Statement

Let KK be a field and let f∈K[x]f \in K[x] be a nonzero polynomial of degree nn. Then ff has at most nn roots in KK. More precisely, the sum of the multiplicities of the roots of ff in KK is at most nn.

Remarks

The proof is an induction on nn with the factor theorem: a root cc gives f=(x−c)gf = (x - c) g with deg⁡g=n−1\deg g = n - 1. The same statement holds over every integral domain. It fails without this hypothesis: in Z/8Z\mathbb{Z}/8\mathbb{Z}, the polynomial x2−1x^2 - 1 has the four roots 1,3,5,71, 3, 5, 7.

Explanations

L1 human

A polynomial of degree 22 over the real numbers has at most two roots. In general, each root takes away one linear factor, and you can do this at most nn times. Strange rings like Z/8Z\mathbb{Z}/8\mathbb{Z} can break this rule, because they have zero divisors.

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To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionPolynomial 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 polynomialdefinitionDegree of a product, statement, no evidence levelDegree of a productstatementMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementRemainder theorem, statement, no evidence levelRemainder theoremstatementFactor theorem, statement, no evidence levelFactor theoremstatementA polynomial over a field has at most as many roots as its degree, statement, no evidence levelA polynomial over a f…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Zero divisor definition
  6. Degree of a polynomial definition
  7. Integral domain definition
  8. Polynomial evaluation definition
  9. Leading coefficient definition
  10. Root of a polynomial definition
  11. Monic polynomial definition
  12. Degree of a product statement
  13. Multiplicity of a root definition
  14. Division algorithm for polynomials statement
  15. Remainder theorem statement
  16. Factor theorem statement
  17. A polynomial over a field has at most as many roots as its degree statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Polynomial, Solving equations

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