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Partial Fraction (Decomposition). Partial Fraction Decomposition of f(x)/d(x). Degree of f ≥ degree of d: Use the division algorithm to divide f by d to obtain the quotient q and remainder r and write f(x) = q(x) + r(x) d(x) d(x)

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Partial fraction decomposition of f x d x
Partial Fraction Decomposition of f(x)/d(x)

  • Degree of f ≥ degree of d: Use the division algorithm to divide f by d to obtain the quotient q and remainder r and write f(x) = q(x) + r(x) d(x) d(x)

  • Factor d(x) into a product of factors of the form (mx + n)u or (ax2 + bx +c)v, where ax2 + bx +c is irreducible.

  • For each factor (mx + n)u: The partial fraction decomposition of r(x)/d(x) must include the sum

    A1 + A2 + … + Au , mx + n (mx + n)2 (mx + n)u

    where A1, A2, …, Au are real numbers.


Continued
continued

  • For each factor (ax2 + bx + c)v: The partial fraction decomposition of r(x)/d(x) must include the sum

    B1x + C1 + B2x + C2 + … + Bvx + Cv ax2 + bx + c (ax2 + bx + c)2 (ax2+bx+c)v

    where B1, B2,…, Bv and C1, C2,…, Cv are real numbers. The partial fraction decomposition of the original rational function is the sum of q(x) and the fraction is the sum of q(x) and the fractions in parts 3 and 4



Repeated linear factors 5x 2 20x 6 x 3 2x 2 x
Repeated linear factors (5x2 + 20x + 6)/(x3+ 2x2 + x)



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