What is Partial Fraction Decomposition?

Partial fraction decomposition is an algebraic process that breaks a complex rational function into a sum of simpler fractions with lower-degree denominators.

When can Partial Fraction Decomposition be applied?

It applies to proper rational functions where numerator polynomial degree is strictly less than denominator polynomial degree. Improper functions require long division first.

What is the Heaviside Cover-Up Method?

The Heaviside cover-up method is a shortcut technique for quickly finding partial fraction numerator constants for distinct non-repeating linear factors.

How do you handle repeated linear factors in partial fractions?

Repeated linear factors (x - a)^m require m separate partial fraction terms with denominators increasing in power from (x - a)^1 up to (x - a)^m.

How do you handle irreducible quadratic factors?

Irreducible quadratic factors (ax^2 + bx + c) receive linear numerators of the form (Ax + B) in the partial fraction expansion.

Why is partial fraction decomposition used in calculus?

It transforms unintegrable rational functions into simple sum terms that can be integrated using basic natural logarithm and inverse tangent formulas.

How are partial fractions used in Laplace Transforms?

Control engineers use partial fractions to break complex transfer functions into standard terms easily converted back to time-domain signals via inverse Laplace tables.

What is an Improper Rational Function?

A rational function P(x)/Q(x) where the degree of numerator P(x) is greater than or equal to the degree of denominator Q(x).

How do you check if a quadratic factor is irreducible?

Calculate the discriminant b^2 - 4ac. If the discriminant is less than zero, the quadratic factor cannot be factored into real linear factors.

What is the connection between partial fractions and logistic growth?

Solving the logistic differential equation requires partial fraction decomposition of 1/[P(K - P)] to integrate and derive the sigmoid population curve.

math calculator

Partial Fractions Decomposition Calculator

Decompose rational expressions into partial fractions with linear, repeated, and quadratic terms.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Denominator Type

Numerator: ax + b

a (x coefficient)

b (constant)

Denominator Roots

Root r₁

Root r₂

Results

Constant A

3.666667

Constant B-0.666667

Decomposed Form

3.6667 / (x − 2) + -0.6667 / (x − -1)

Step-by-Step Algebraic Proof

Set up the decomposition for \(\frac{3x +5}{(x-2)(x+1)}\):

\[\frac{3x +5}{(x - 2)(x - -1)} = \frac{A}{x - 2} + \frac{B}{x - -1}\]

Multiply both sides by \((x - 2)(x - -1)\):

\[3x +5 = A(x - -1) + B(x - 2)\]

Substitute \(x = 2\) to solve for A:

\[3(2) + 5 = A(2 - -1) \implies 11 = A(3) \implies A = \frac{11}{3} = 3.6667\]

Substitute \(x = -1\) to solve for B:

\[3(-1) + 5 = B(-1 - 2) \implies 2 = B(-3) \implies B = \frac{2}{-3} = -0.6667\]

Final result:

\[\frac{3x +5}{(x - 2)(x - -1)} = \frac{3.667}{x - 2} + \frac{-0.667}{x - -1}\]

Integration result: \[\int \frac{3x +5}{(x - 2)(x - -1)}\,dx = 3.667\ln|x - 2| + -0.667\ln|x - -1| + C\]

Formula

Partial Fraction Decomposition Formulas

\frac{P(x)}{(x-r_1)(x-r_2)} = \frac{A}{x-r_1} + \frac{B}{x-r_2}, \quad A = \frac{P(r_1)}{r_1-r_2}

Decompose a rational expression into a sum of simpler fractions. Each partial fraction can be directly integrated using the natural logarithm formula.

Step by step

How the calculation works

1Select distinct linear factors or repeated factors mode.
2Enter numerator coefficients (ax + b) and the roots of the denominator factors.
3The calculator solves for A and B constants using the root-substitution method.

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Fraction Calc

Fraction Calculator

Add, subtract, and simplify fractions step-by-step.

Quadratic Solver

Find roots of ax² + bx + c = 0 for use as denominator factors.

math calculator

Partial Fractions Decomposition Calculator

Decompose rational expressions into partial fractions with linear, repeated, and quadratic terms.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Denominator Type

Numerator: ax + b

a (x coefficient)

b (constant)

Denominator Roots

Root r₁

Root r₂

Results

Constant A

3.666667

Constant B-0.666667

Decomposed Form

3.6667 / (x − 2) + -0.6667 / (x − -1)

Step-by-Step Algebraic Proof

Set up the decomposition for \(\frac{3x +5}{(x-2)(x+1)}\):

\[\frac{3x +5}{(x - 2)(x - -1)} = \frac{A}{x - 2} + \frac{B}{x - -1}\]

Multiply both sides by \((x - 2)(x - -1)\):

\[3x +5 = A(x - -1) + B(x - 2)\]

Substitute \(x = 2\) to solve for A:

\[3(2) + 5 = A(2 - -1) \implies 11 = A(3) \implies A = \frac{11}{3} = 3.6667\]

Substitute \(x = -1\) to solve for B:

\[3(-1) + 5 = B(-1 - 2) \implies 2 = B(-3) \implies B = \frac{2}{-3} = -0.6667\]

Final result:

\[\frac{3x +5}{(x - 2)(x - -1)} = \frac{3.667}{x - 2} + \frac{-0.667}{x - -1}\]

Integration result: \[\int \frac{3x +5}{(x - 2)(x - -1)}\,dx = 3.667\ln|x - 2| + -0.667\ln|x - -1| + C\]

Formula

Partial Fraction Decomposition Formulas

\frac{P(x)}{(x-r_1)(x-r_2)} = \frac{A}{x-r_1} + \frac{B}{x-r_2}, \quad A = \frac{P(r_1)}{r_1-r_2}

Decompose a rational expression into a sum of simpler fractions. Each partial fraction can be directly integrated using the natural logarithm formula.

Step by step

How the calculation works

1Select distinct linear factors or repeated factors mode.
2Enter numerator coefficients (ax + b) and the roots of the denominator factors.
3The calculator solves for A and B constants using the root-substitution method.

Related guides

Learn more about this calculation

Guide

Find roots of ax² + bx + c = 0 for use as denominator factors.