Partial Decomposition Calculator

Write it expanded or as a product of factors, e.g. (x - 3)(x + 2).
Partial fraction decomposition
-1/(x + 2) + 4/(x - 3)
Partial Fraction Decomposition Calculator
Polynomial part (from long division)
0
Factored denominator
(x + 2)(x - 3)
Degree of the denominator
2
The input fraction was proper
yes
Largest check error at sample points
0
Worked out from James Stewart, Calculus: Early Transcendentals, section 7.4: Integration of Rational Functions by Partial Fractions and 1 other source.

How this is worked out

6 steps
  1. N(x) = 3x + 11, D(x) = x^2 - x - 6 (degree 2).
  2. The fraction is already proper, so no long division is needed.
  3. Factor the denominator: (x + 2)(x - 3)
  4. Write one unknown per power of each factor (2 unknowns for a degree-2 denominator).
  5. Clear denominators and match coefficients of x^0 ... x^1, then solve the 2 x 2 system.
  6. Decomposition: -1/(x + 2) + 4/(x - 3)
Termstable
denominatornumeratorterm
(x + 2)-1-1/(x + 2)
(x - 3)44/(x - 3)
Coefficients in term ordertable
-1, 4

Results are general information, not financial, tax, legal, medical or engineering advice.