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Partial Fraction Decomposition

Break P(x)/Q(x) into simpler fractions. Distinct linear: A/(x-r₁)+B/(x-r₂). Repeated: A/(x-r)+B/(x-r)². Quadratic: (Ax+B)/(x²+c). Cover-up method for constants.

Concept Fundamentals
A/(x-r₁)+B/(x-r₂)
Distinct
A/(x-r)+B/(x-r)²
Repeated
(Ax+B)/(x²+c)
Quadratic
Method
Cover-up

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Cover-up: for A/(x-r₁), substitute x=r₁ after covering (x-r₁). ∫A/(x-r)dx = A·ln|x-r|. ∫(Ax+B)/(x²+1) uses arctan. Proper fraction first: deg(P) < deg(Q).

Key quantities
A/(x-r₁)+B/(x-r₂)
Distinct
Key relation
A/(x-r)+B/(x-r)²
Repeated
Key relation
(Ax+B)/(x²+c)
Quadratic
Key relation
Method
Cover-up
Key relation

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Why: Partial fractions enable integration of rational functions. Each term has a standard antiderivative. Laplace transforms: decompose Y(s) for inverse.

How: Set up form based on denominator factors. Multiply through, equate coefficients or use cover-up. Distinct linear: substitute roots. Repeated: equate coefficients.

Cover-up: for A/(x-r₁), substitute x=r₁ after covering (x-r₁).∫A/(x-r)dx = A·ln|x-r|. ∫(Ax+B)/(x²+1) uses arctan.
Sources:Khan AcademyNCTM Standards

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Decompose Partial FractionsEnter rational function

📐 Examples — Click to Load

Decomposition Type

partial_fraction_decompose.sh
DECOMPOSED
$ decompose --expression="(3x+5)/((x-1)(x+2))"
Original
(3x+5)/((x-1)(x+2))
Decomposition
2.6666666666666665/(x-1) + 0.3333333333333333/(x+2)
A
2.6667
B
0.3333
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Partial Fraction Constants

Contribution of Each Fraction

📐 Calculation Steps

Original: (3x+5)/((x-1)(x+2))
Form: A/(x-1) + B/(x+2)
Multiply by (x-1)(x--2): 3x+5 = A(x--2) + B(x-1)
At x=1: 3(1)+5 = A(1--2) ⇒ A = 2.6667
At x=-2: 3(-2)+5 = B(-2-1) ⇒ B = 0.3333
Decomposition: 2.6666666666666665/(x-1) + 0.3333333333333333/(x+2)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

∫

Partial fractions enable integration

— Calculus

📐

Cover-up method for constants

— Algebra

📋 Key Takeaways

  • • Partial fraction decomposition breaks a rational function into simpler fractions
  • • Distinct linear factors: (ax+b)/((x-r₁)(x-r₂)) = A/(x-r₁) + B/(x-r₂)
  • • Repeated roots: (ax+b)/(x-r)² = A/(x-r) + B/(x-r)²
  • • Irreducible quadratics: Use form (Ax+B)/(ax²+bx+c)
  • • Cover-up method: Substitute roots to find constants quickly

💡 Why Partial Fractions Matter

∫Integration: ∫(1/(x-1) + 4/(x+2))dx = ln|x-1| + 4ln|x+2| + C. Each term integrates easily.Source: Calculus
⚡Laplace transforms: Decompose Y(s) to find inverse transform y(t) using known pairs.Source: Differential Eqs
📐Cover-up method: For A/(x-r₁), substitute x=r₁ and "cover up" (x-r₁) in the denominator.Source: Algebra
🔢Method of undetermined coefficients: Set up form, multiply through, equate coefficients.Source: Linear Algebra
🏗️Control systems: Transfer functions are rational; partial fractions help analyze stability.Source: Engineering
📊Proper fraction: Degree(numerator) &lt; Degree(denominator). Do polynomial division first if improper.Source: Precalculus

📖 Method of Undetermined Coefficients

For (3x+5)/((x-1)(x+2)) = A/(x-1) + B/(x+2): Multiply both sides by (x-1)(x+2) to get 3x+5 = A(x+2) + B(x-1). Substitute x=1: 8=3A ⇒ A=8/3. Substitute x=-2: -1=-3B ⇒ B=1/3.

Cover-Up Method

To find A for A/(x-1): "Cover up" (x-1) in the original, substitute x=1 into what remains: A = (3(1)+5)/(1+2) = 8/3.

📊 Decomposition Forms

DenominatorPartial Fraction Form
(x-r₁)(x-r₂)A/(x-r₁) + B/(x-r₂)
(x-r)²A/(x-r) + B/(x-r)²
x²+1 (irreducible)(Ax+B)/(x²+1)
(x-r)(x²+1)A/(x-r) + (Bx+C)/(x²+1)

🎯 Expert Tips

💡 Check Proper First

If deg(P) ≥ deg(Q), do polynomial long division first. Decompose only the remainder over Q.

💡 Verify Your Answer

Add the partial fractions back together. You should get the original rational expression.

💡 Laplace Applications

For L⁻¹Y(s), decompose Y(s) into terms like 1/(s-a) whose inverse is e^(at).

💡 Integration Result

∫A/(x-r)dx = A·ln|x-r|. ∫(Ax+B)/(x²+1)dx uses arctan. Each term has a standard antiderivative.

❓ FAQ

What is partial fraction decomposition?

Breaking a rational function P(x)/Q(x) into a sum of simpler fractions whose denominators are factors of Q(x). Essential for integration and Laplace transforms.

When do I need polynomial division first?

When the degree of the numerator is greater than or equal to the degree of the denominator. The result is polynomial + proper fraction.

What is the cover-up method?

For distinct linear factors: to find A in A/(x-r), substitute x=r into the original after "covering" (x-r). A = P(r)/Q(r) where Q is the product of other factors evaluated at r.

How do I handle repeated roots?

For (x-r)ⁿ, include terms A₁/(x-r) + A₂/(x-r)² + ... + Aₙ/(x-r)ⁿ. Substitute x=r for the highest power term, then equate coefficients for others.

What about irreducible quadratics?

Use (Ax+B)/(ax²+bx+c). Multiply through, expand, and equate coefficients of 1, x, x² to get a system of equations.

Why is this useful in calculus?

∫1/(x-a)dx = ln|x-a|. ∫1/(x²+a²)dx = (1/a)arctan(x/a). Decomposition turns hard integrals into sums of easy ones.

⚠️ Disclaimer: This calculator handles common decomposition types. For complex denominators or improper fractions, manual polynomial division may be required first. Educational use only.

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