Partial fraction decomposition is a method for rewriting a complicated rational expression as a sum of simpler fractions. It matters in calculus because many integrals are much easier to evaluate after this rewrite. The idea is especially useful when the denominator factors into linear or irreducible quadratic pieces.
It also helps in differential equations, Laplace transforms, and algebraic simplification.
The method starts by factoring the denominator completely and then choosing a fraction form for each factor. Unknown constants are placed in the numerators, and those constants are found by clearing denominators and matching coefficients or substituting convenient values. For distinct linear factors, the numerators are constants such as A or B.
For repeated factors and quadratic factors, the setup changes in a predictable way, so choosing the correct form is a key step.
Understanding Partial Fraction Decomposition
Before starting the decomposition, check whether the numerator has degree at least as large as the denominator. If it does, polynomial division must come first. The result is a polynomial plus a proper rational fraction.
Only the proper fraction needs partial fractions. Skipping this step gives a setup that cannot work. Factoring also needs care.
A quadratic that has no real roots stays as one quadratic factor. A quadratic with real roots should be split into its linear factors. The factorization determines every term that must appear later.
The numerator chosen for each partial fraction comes from the type of denominator factor. A linear factor gets a constant because there is no smaller nonconstant polynomial degree available. A quadratic factor gets a linear numerator, with an x term and a constant term.
This gives enough flexibility to reproduce the original numerator after denominators are cleared. Repeated factors need a term for every power from the first power up to the highest power.
For example, a cube of one linear factor requires three separate fractions. Missing one of these terms is a common error, even when its final coefficient happens to be zero.
After multiplying by the full denominator, the equation becomes a polynomial identity. This means the two expanded sides must agree for every allowed value of x. Substituting a value that makes one factor zero can isolate a coefficient quickly.
This method is especially useful for distinct linear factors. Coefficient matching is more reliable when several unknowns remain or when repeated factors are present. Group the constant terms, x terms, squared terms, and higher powers carefully.
Then compare each pair of coefficients. A useful check is to substitute an ordinary value of x into both the original fraction and the completed decomposition. Agreement at several values helps catch sign errors.
Integration is the reason the decomposition is worth the effort. A constant divided by a linear expression produces a logarithm of the absolute value of that linear expression. A term with a repeated linear factor usually uses the power rule and gives a negative power.
Quadratic terms require more thought. Sometimes the linear numerator can be split into one part related to the derivative of the quadratic denominator and one leftover constant part. The derivative-related part gives a logarithm.
For a quadratic like x squared plus a positive constant, the leftover part leads to an inverse tangent result. Keep the original denominator restrictions in mind. The antiderivative applies separately on intervals that do not cross values where the original rational function is undefined.
Key Facts
- For distinct linear factors, , provided degree of is less than degree of denominator.
- Example: .
- Clear denominators by multiplying both sides by the full denominator, then solve for constants.
- Coefficient matching means expanding and comparing powers of x on both sides.
- For a repeated linear factor, use separate terms: .
- For an irreducible quadratic factor, use a linear numerator: .
Vocabulary
- Rational function
- A rational function is a quotient of two polynomials, such as P(x)/Q(x).
- Partial fraction decomposition
- Partial fraction decomposition is the process of rewriting a rational function as a sum of simpler fractions.
- Linear factor
- A linear factor is a first degree factor of the form or .
- Repeated factor
- A repeated factor is a denominator factor that appears more than once, such as .
- Coefficient matching
- Coefficient matching is solving for unknown constants by making the coefficients of like powers of x equal on both sides.
Common Mistakes to Avoid
- Using partial fractions before checking degrees, which is wrong because an improper rational function must first be divided so the numerator degree is less than the denominator degree.
- Forgetting to factor the denominator completely, which is wrong because the decomposition form depends on the actual factors, not the unfactored polynomial.
- Using only one term for a repeated factor, which is wrong because a factor like requires both and .
- Putting a constant numerator over an irreducible quadratic, which is wrong because a factor like needs a linear numerator .
Practice Questions
- 1 Decompose (7x + 1)/[(x - 2)(x + 3)] into partial fractions and solve for the constants.
- 2 Decompose into partial fractions. Write the correct form first, then find the constants.
- 3 Explain why the decomposition of must use the form instead of .