Partial fraction decomposition is a method for rewriting one complicated rational expression as a sum of simpler fractions. It matters because many algebra, calculus, and differential equations problems become easier after the expression is split apart. The method is especially useful for integrating rational functions, finding inverse Laplace transforms, and simplifying algebraic work.
It turns a single dense fraction into pieces whose denominators are easier to understand.
Key Facts
- Use partial fractions only after the rational expression is proper: degree of numerator < degree of denominator.
- If the fraction is improper, divide first: P(x)/Q(x) = quotient + remainder/Q(x).
- For a distinct linear factor (x - a), use A/(x - a).
- For a repeated linear factor (x - a)^n, use A1/(x - a) + A2/(x - a)^2 + ... + An/(x - a)^n.
- For an irreducible quadratic factor ax^2 + bx + c, use (Ax + B)/(ax^2 + bx + c).
- After multiplying by the common denominator, solve for constants by substituting convenient x-values or matching coefficients.
Vocabulary
- Rational expression
- A fraction whose numerator and denominator are polynomials.
- Proper rational expression
- A rational expression in which the degree of the numerator is less than the degree of the denominator.
- Linear factor
- A polynomial factor of degree 1, such as x - 3 or 2x + 5.
- Irreducible quadratic
- A quadratic factor that cannot be factored into real linear factors.
- Coefficient matching
- A method of solving for unknown constants by setting coefficients of equal powers of x equal on both sides of an identity.
Common Mistakes to Avoid
- Skipping polynomial division for an improper fraction, which is wrong because partial fractions are set up only for the proper remainder part.
- Using only one term for a repeated factor, which is wrong because (x - a)^n requires fractions with every power from 1 through n.
- Putting a constant numerator over a quadratic factor, which is wrong because an irreducible quadratic needs a linear numerator Ax + B.
- Solving the equation only at one x-value, which is wrong because each unknown constant needs enough independent information to be determined.
Practice Questions
- 1 Decompose 5/(x^2 - 1) into partial fractions.
- 2 Decompose (3x + 7)/((x - 2)(x + 1)) into partial fractions.
- 3 Explain why (2x + 1)/((x - 3)^2(x^2 + 4)) must be set up with terms A/(x - 3) + B/(x - 3)^2 + (Cx + D)/(x^2 + 4).