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  2. Multiplicity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Multiplicity_(mathematics)

    This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra.

  3. Hilbert–Samuel function - Wikipedia

    en.wikipedia.org/wiki/Hilbert–Samuel_function

    This integer is called the multiplicity of the ideal . When I = m {\displaystyle I=m} is the maximal ideal of A {\displaystyle A} , one also says e {\displaystyle e} is the multiplicity of the local ring A {\displaystyle A} .

  4. Polynomial expansion - Wikipedia

    en.wikipedia.org/wiki/Polynomial_expansion

    In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...

  5. Polynomial root-finding - Wikipedia

    en.wikipedia.org/wiki/Polynomial_root-finding...

    So, except for very low degrees, root finding of polynomials consists of finding approximations of the roots. By the fundamental theorem of algebra, a polynomial of degree n has exactly n real or complex roots counting multiplicities. It follows that the problem of root finding for polynomials may be split in three different subproblems;

  6. File:Polynomial roots multiplicity.svg - Wikipedia

    en.wikipedia.org/wiki/File:Polynomial_roots...

    English: Graph of the polynomial y = x^3 + 2*x^2 - 7*x + 4 with its roots (zeros) -4 and 1 marked. The root -4 is a 'simple' root (of multiplicity 1), and therefore the graph crosses the x-axis at this root. The root 1 is of even multiplicity and therefore the graph bounces off the x-axis at this root.

  7. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    Any nth degree polynomial has exactly n roots in the complex plane, if counted according to multiplicity. So if f(x) is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the minimum number of nonreal roots is equal to (+),

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