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