AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Donate Contacts

← Back to list of members

SolveByReducingThePolynomialOverALinearFactor

 Method with 2 overloads

• SolveByReducingThePolynomialOverALinearFactor​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean,​System.​Boolean)

  Summary

  SolveByReducingThePolynomialOverALinearFactor(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean),
  or with alone a polynomial over a power of one linear and nothing else,
  that linear with a symbol in it: t^9/(a + b t)^8 is a polynomial and eight powers of
  1/(a + b t), each the table's. SolveByPartialFractions(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) asks it so before
  it divides: divided out and decomposed with the symbols in it, that one went round the
  substitution search a dozen levels deep and took forty-six seconds, and with numbers in
  place of a and b it took sixty milliseconds.
  

• SolveByReducingThePolynomialOverALinearFactor​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

  Summary

  A polynomial numerator over a denominator with a power of a linear in it beside a
  factor that is not a polynomial -- P(x)/((a + b x)^k S(x)) -- with the
  polynomial written in powers of the linear at its root: what the power divides goes
  over S alone, and each lower power over its own power of the linear.
  

  Remarks

  P(x) = p_0 + p_1 (a + b x) + ... + (a + b x)^k Q(x) is a polynomial identity,
  its coefficients the Taylor coefficients of P at -a/b over the powers
  of b, so the integrand is Q/S + sum p_j /((a + b x)^(k-j) S) exactly and
  everywhere. (A + B x + C x^2 + D x^3)/((a + b x) sqrt(c + d x)) was answered
  through the substitution u = sqrt(c + d x) term by term, a cubic in
  u^2 - c over a symbolic quadratic in u each time, in a hundred kilobytes
  of piecewise that did not evaluate within the corpus's budget; reduced, it is a
  quadratic over the root, three powers, and one p_0/((a + b x) sqrt(c + d x)).
  Only beside a factor that is not a polynomial in x: a quotient of polynomials
  is the partial fractions' and they do this and more.
  https://github.com/asc-community/AngouriMath/issues/718