Proper and Improper Fractions
Introduction
In partial fractions, a rational fraction is classified as proper or improper by comparing the degrees of its numerator and denominator.
Before learning how to resolve partial fractions, it is important to understand this difference because improper rational fractions usually need to be simplified using polynomial division before applying partial fraction decomposition.
1. Proper Rational Fraction
A rational fraction is called a proper fraction when the degree of the numerator is less than the degree of the denominator.
Example 1:
(2x + 3) / (x² + 1)
The numerator has degree 1, and the denominator has degree 2.
Since 1 < 2, this is a proper rational fraction.
Example 2:
(x + 5) / (x³ + 2x)
The numerator has degree 1, and the denominator has degree 3.
Since 1 < 3, this is also a proper rational fraction.
2. Improper Rational Fraction
A rational fraction is called an improper fraction when the degree of the numerator is greater than or equal to the degree of the denominator.
Example 1:
(x² + 3) / (x + 1)
The numerator has degree 2, and the denominator has degree 1.
Since 2 > 1, this is an improper rational fraction.
Example 2:
(2x² + 5) / (x² + 1)
Both the numerator and denominator have degree 2.
Since their degrees are equal, this is also an improper rational fraction.
3. Comparison Between Proper and Improper Fractions
Feature | Proper Fraction | Improper Fraction |
|---|---|---|
Degree of numerator | Less than denominator | Greater than or equal to denominator |
Example | (x + 2) / (x² + 1) | (x² + 1) / (x + 2) |
Polynomial division required first? | Usually not | Yes, before partial fraction decomposition |
4. Solved Examples
Example 1 — Basic Level
Classify the following fraction:
(x + 2) / (x² + 4)
Step 1: Degree of numerator = 1.
Step 2: Degree of denominator = 2.
Step 3: Compare the degrees.
Since 1 < 2, the fraction is a proper rational fraction.
Example 2 — Medium Level
Classify the following fraction:
(3x² + 2x + 1) / (x² − 5)
Step 1: Degree of numerator = 2.
Step 2: Degree of denominator = 2.
Step 3: The degrees are equal.
Therefore, the fraction is an improper rational fraction.
Example 3 — Challenging Level
Classify the following fraction:
(2x³ + x² − 4) / (x² + 3x + 1)
Step 1: Degree of numerator = 3.
Step 2: Degree of denominator = 2.
Step 3: Since 3 > 2, the fraction is an improper rational fraction.
For partial fraction decomposition, polynomial division must be performed first.
5. Why Is This Classification Important?
Consider the fraction:
(x² + 1) / (x + 1)
The numerator has degree 2, while the denominator has degree 1. Therefore, it is improper.
Divide x² + 1 by x + 1 using polynomial division:
x² + 1 = (x + 1)(x − 1) + 2
Dividing both sides by x + 1 gives:
(x² + 1) / (x + 1) = x − 1 + 2 / (x + 1)
The original improper fraction has now been expressed as a polynomial plus a proper rational fraction.
This is the form needed before proceeding with partial fraction decomposition.
Key Takeaways
A proper rational fraction has a numerator degree lower than its denominator degree.
An improper rational fraction has a numerator degree greater than or equal to its denominator degree.
Compare the highest powers of the variable to determine the degrees.
Perform polynomial division before applying partial fractions to an improper rational fraction.
In summary, identifying proper and improper rational fractions is an essential first step in solving partial fraction problems. Once you can compare polynomial degrees, you can determine whether polynomial division is needed.