Partial fraction decomposition
The calculator decomposes a polynomial fraction to several fractions with a simpler denominator.
The calculator below transforms a polynomial fraction into a sum of simpler fractions. The fraction numerator is defined by a sequence of coefficients (starting from higherdegree coefficient to lower one). The denominator is given by a product of linear or quadratic polynomials raised to a degree >=1.
Denominator polynomial factors
arrow_upwardarrow_downwardFactor  arrow_upwardarrow_downwardExponent  

Solution
The following calculator provides a 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.
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)
 if the degree of P_{1}(x) is greater than or equal to the degree of Q_{1}(x), do the long division to find the common polynomial term (quotient) and the new numerator P_{2}(x) (remainder), which degree is less than Q_{1}(x) degree:
, where
 find the denominator factorization as l linear factors for real roots of Q_{1}(x) and n quadratic factors for complex roots of Q_{1}(x):
 then the partial fraction decomposition takes the form:
, where a_{jk}, b_{jk},c_{jk} 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 a_{jk}, b_{jk},c_{jk}
 equate each coefficient of P_{2}(x) to the linear expression with a_{jk}, b_{jk},c_{jk} corresponding to the same degree of x
 create and solve the system of linear equations to obtain a_{jk}, b_{jk},c_{jk}
You may switch on the 'Show details' toggle of the calculators above to study the procedure steps using an example.

V.A.Zorich Math analysis vol.1 ↩
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