Partial fraction decomposition

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 higher-degree coefficient to lower one). The denominator is given by a product of linear or quadratic polynomials raised to a degree >=1.

Partial fraction decomposition

Numerator polynomial

Space separated polynomial coefficients.



Denominator polynomial factors

	Factor	Exponent
			

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Calculate

Calculation precision

Exact

Rounded

Digits after the decimal point: 2

Problem

 

Solution

Solution

 

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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 decomposition 2

Numerator polynomial

Space separated polynomial coefficients.

Denominator polynomial

Space separated polynomial coefficients.

Show details

Calculate

Calculation precision

Exact

Rounded

Digits after the decimal point: 2

Problem

 

Solution

 

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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 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:

, where

• find the denominator factorization as l linear factors for real roots of Q₁(x) and n quadratic factors for complex roots of Q₁(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 P₂(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.