Partial fractions decomposition rewrites a complicated rational expression as a sum of simpler rational expressions. Students need this cheat sheet when integrating rational functions, simplifying algebraic expressions, and preparing for precalculus or calculus. It helps identify which partial fraction form matches each type of denominator factor.
It also shows how to set up equations before solving for unknown coefficients.
The main idea is to factor the denominator completely, then assign the correct numerator form to each factor. A linear factor such as gets a constant numerator, while an irreducible quadratic such as gets a linear numerator. If the rational expression is improper, divide first so the numerator degree is less than the denominator degree.
After setup, multiply by the common denominator and solve for the unknown constants.
Key Facts
- A rational expression is proper when , and partial fractions should be set up only after this is true.
- For a distinct linear factor , use the form .
- For repeated linear factors , use the full chain .
- For an irreducible quadratic factor , use the form .
- For repeated irreducible quadratic factors , include .
- After writing the partial fraction form, multiply both sides by the least common denominator to create a polynomial identity.
- Coefficients can be found by substituting convenient values of or by matching coefficients of equal powers of .
- If , use polynomial division first: with .
Vocabulary
- Rational expression
- A quotient of two polynomials, written as where .
- Proper rational expression
- A rational expression in which the numerator has smaller degree than the denominator, so .
- Linear factor
- A first-degree factor of a polynomial, such as or .
- Irreducible quadratic
- A quadratic factor that cannot be factored into real linear factors.
- Repeated factor
- A factor that appears with an exponent greater than , such as .
- Polynomial identity
- An equation between polynomials that is true for all allowed values of .
Common Mistakes to Avoid
- Skipping polynomial division when the fraction is improper is wrong because partial fractions require before decomposition.
- Using only one term for a repeated factor is wrong because needs every power from through , such as for .
- Putting a constant numerator over an irreducible quadratic is wrong because needs a linear numerator .
- Canceling factors before checking the domain can be wrong because the original expression still has excluded values where the denominator equals .
- Substituting values that make a denominator zero in the original equation is wrong unless you have already multiplied through to form a valid polynomial identity.
Practice Questions
- 1 Set up the partial fraction form for .
- 2 Decompose into partial fractions.
- 3 Set up the partial fraction form for .
- 4 Explain why is not the most general partial fraction form for an irreducible quadratic factor.
Understanding Partial Fractions Decomposition Reference
Partial fractions works because each denominator factor creates its own kind of remainder. When a polynomial is divided by a linear factor, the remainder is a constant. When it is divided by a quadratic factor that cannot be factored using real numbers, the remainder can have degree one.
It therefore needs a constant term plus a variable term. This is more than a rule to memorize. It comes from polynomial division.
A numerator with too much degree can hide part of the expression in a polynomial quotient. Removing that quotient first makes the remaining pieces uniquely determined.
Repeated factors need extra care because each power can contribute a different part of the final expression. A factor raised to a higher power does not replace the lower powers. Think of the separate fractions as slots that allow enough freedom to rebuild the original numerator after denominators are cleared.
If one slot is missing, the equations may have no solution even when the setup looks close. This is a common source of errors.
Students often notice the repeated factor but write only the highest power. Listing every power in order prevents that mistake.
After multiplying through by the full denominator, the result is a polynomial identity. It must be true for every allowed value of the variable. This gives two useful solving methods.
For distinct linear factors, choosing a value that makes one factor zero can make most terms disappear. The remaining term often reveals one unknown coefficient quickly. For quadratic factors, repeated factors, or complicated expressions, matching coefficients is more reliable.
Expand both sides carefully, group terms with the same power of the variable, then set their coefficients equal. Using a few test values can check the work, but test values alone may not provide enough independent equations.
The method becomes especially useful in calculus because the simpler pieces have familiar antiderivatives. A linear denominator often leads to a logarithm. A quadratic written as a square plus a constant can lead to an inverse tangent form.
Repeated factors create terms with negative powers that use the power rule. In algebra, the most important habit is verification. Combine the completed fractions over the original common denominator and simplify the numerator.
It should match the starting numerator exactly. Watch for sign errors when subtracting expressions, missing parentheses during expansion, and factors that were not fully factored before the setup began.