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  2. Factorization of polynomials over finite fields - Wikipedia

    en.wikipedia.org/wiki/Factorization_of...

    The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2, P = (x + 1) 4. On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have; If =, then = (+) ().

  3. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...

  4. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_decomposition

    In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

  5. Factorization of polynomials - Wikipedia

    en.wikipedia.org/wiki/Factorization_of_polynomials

    The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. [1] Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra ...

  6. Primitive polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Primitive_polynomial...

    The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to 3 21 = 8 are 1, 3, 5, and 7, the four primitive roots in GF(3 2) are α, α 3 = 2α + 1, α 5 = 2α, and α 7 = α + 2. The primitive roots α and α 3 are algebraically conjugate.

  7. Winners and losers of 4 Nations Face-Off: Canada beats USA ...

    www.aol.com/winners-losers-4-nations-face...

    Games were tight — even a 6-1 U.S. win over Finland was 2-1 heading into the third period — and intense. Three games went to overtime, and the NHL's top player, three-time MVP Connor McDavid ...

  8. Irreducible polynomial - Wikipedia

    en.wikipedia.org/wiki/Irreducible_polynomial

    In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.

  9. Worst Housing Markets for Growth in 16 East Coast States - AOL

    www.aol.com/finance/worst-housing-markets-growth...

    In final calculations factors (1), (2), (9), and (10) were weighted 2x and factor (5) was weighted 0.5x. All data was collected on and is up to date as of Jan. 29, 2025. More From GOBankingRates.