Partial fraction decomposition

The calculator decomposes a polynomial fraction to several fractions with a simpler denominator.

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The calculator below transforms a polynomial fraction to a sum of simpler fractions. The fraction numerator is defined by a sequence of coefficients (starting form higher-degree coefficient to lower one). The denominator is given by a product of linear or quadratic polynomials raised to a degree >=1.

PLANETCALC, Partial fraction decomposition

Partial fraction decomposition

Space separated polynomial coefficients.

Denominator polynomial factors

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The following calculator provides simpler method to input the denominator and more complicated logic to find the fraction decomposition. But this calculator will not work if the denominator polynomial has irreducible factors of degree>2 in rational numbers.

PLANETCALC, Partial fraction decomposition 2

Partial fraction decomposition 2

Space separated polynomial coefficients.
Space separated polynomial coefficients.

Partial fraction expansion procedure

The partial fraction decomposition procedure of a polynomial fraction P(x)/Q(x) is as follows:

  • convert the denominator polynomial to monic by dividing P (x) and Q (x) by the leading coefficient of Q (x)

\frac{P_1(x)}{Q_1(x)} = \frac{P(x)/lc(Q(x)}{Q(x)/lc(Q(x))}

  • if the degree of P1(x) is greater than or equal to the degree of Q1(x), do the long division to find the common polynomial term (quotient) and the new numerator P2(x) (remainder), which degree is less than Q1(x) degree:

\frac{P_1(x)}{Q_1(x)} = quot(P_1(x),Q_1(x))  + \frac{P_2(x)}{Q_1(x)}, where P_2(x)=\frac{rem(P_1(x),Q_1(x))}{Q_1(x)}

  • find the denominator factorization as l linear factors for real roots of Q1(x) and n quadratic factors for complex roots of Q1(x):

Q_1(x) = (x-x_1)^{k_1}\cdots(x-x_l)^{k_l}(x^2+p_1x+q_1)^{m_1}\cdots(x^2+p_nx+q_n)^{m_n}

  • then the partial fraction decomposition takes the form:

\frac{P_2(x)}{Q_1(x)} =  \sum_{j=1}^l\sum_{k=1}^{k_j} \frac{a_{jk}}{(x-x_j)^k} + \sum_{j=1}^n\sum_{k=1}^{m_j} \frac{b_{jk}x+c_{jk}}{(x^2+p_jx+q_j)^k}, where ajk, bjk,cjk are real numbers. 1

  • reduce the right side numerator to a common denominator
  • expand the numerator polynomial factors and express the numerator polynomial coefficients in terms of linear expression of unknown constants ajk, bjk,cjk
  • equate each coefficient of P2(x) to the linear expression with ajk, bjk,cjk corresponding to the same degree of x
  • create and solve the system of linear equations to obtain ajk, bjk,cjk
    You may switch on the 'Show details' toggle of the calculators above to study the procedure steps using an example.

  1. V.A.Zorich Math analysis vol.1 

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