Mathwords logoReference LibraryMathwords

Proper Rational Expression

Proper Rational Expression

A rational expression in which the degree of the numerator polynomial is less than the degree of the denominator polynomial.

 

Example: (14x − 3) / (5x² + 2x − 3)

 

 

 

See also

Improper rational expression

Key Formula

P(x)Q(x)is proper whendeg⁡(P)<deg⁡(Q)\frac{P(x)}{Q(x)} \quad \text{is proper when} \quad \deg(P) < \deg(Q)
Where:
  • P(x)P(x) = The numerator polynomial
  • Q(x)Q(x) = The denominator polynomial, where Q(x) ≠ 0
  • deg⁡(P)\deg(P) = The degree of the numerator polynomial (its highest power of x)
  • deg⁡(Q)\deg(Q) = The degree of the denominator polynomial (its highest power of x)

Worked Example

Problem: Determine whether the rational expression (3x² + 5x − 2) / (x³ + 4x) is proper or improper.
Step 1: Identify the numerator polynomial and find its degree. The highest power of x in the numerator is x², so the degree is 2.
P(x)=3x2+5x−2⇒deg⁡(P)=2P(x) = 3x^2 + 5x - 2 \quad \Rightarrow \quad \deg(P) = 2
Step 2: Identify the denominator polynomial and find its degree. The highest power of x in the denominator is x³, so the degree is 3.
Q(x)=x3+4x⇒deg⁡(Q)=3Q(x) = x^3 + 4x \quad \Rightarrow \quad \deg(Q) = 3
Step 3: Compare the two degrees. Since 2 < 3, the degree of the numerator is less than the degree of the denominator.
deg⁡(P)=2<3=deg⁡(Q)\deg(P) = 2 < 3 = \deg(Q)
Answer: The expression (3x² + 5x − 2) / (x³ + 4x) is a proper rational expression because the numerator's degree (2) is less than the denominator's degree (3).

Another Example

This example shows how an improper rational expression can be converted into a polynomial plus a proper rational expression using long division — a key technique in algebra and calculus.

Problem: An improper rational expression (x³ + 2x + 7) / (x² + 1) is given. Use polynomial long division to rewrite it as a polynomial plus a proper rational expression.
Step 1: Check degrees: the numerator has degree 3 and the denominator has degree 2. Since 3 ≥ 2, this is improper. We need to divide.
deg⁡(x3+2x+7)=3≥2=deg⁡(x2+1)\deg(x^3 + 2x + 7) = 3 \geq 2 = \deg(x^2 + 1)
Step 2: Perform polynomial long division. Divide x³ + 2x + 7 by x² + 1. The first term is x³ ÷ x² = x. Multiply: x(x² + 1) = x³ + x. Subtract from the numerator.
(x3+2x+7)−(x3+x)=x+7(x^3 + 2x + 7) - (x^3 + x) = x + 7
Step 3: The remainder is x + 7, which has degree 1. Since 1 < 2 (the denominator's degree), the remainder over the divisor forms a proper rational expression.
x3+2x+7x2+1=x+x+7x2+1\frac{x^3 + 2x + 7}{x^2 + 1} = x + \frac{x + 7}{x^2 + 1}
Step 4: Verify: the fraction (x + 7)/(x² + 1) is proper because deg(x + 7) = 1 < 2 = deg(x² + 1).
x+7x2+1is proper since 1<2\frac{x + 7}{x^2 + 1} \quad \text{is proper since } 1 < 2
Answer: The improper expression rewrites as x + (x + 7)/(x² + 1), where (x + 7)/(x² + 1) is a proper rational expression.

Frequently Asked Questions

What is the difference between a proper and improper rational expression?
A proper rational expression has the degree of its numerator strictly less than the degree of its denominator. An improper rational expression has the numerator's degree greater than or equal to the denominator's degree. You can always convert an improper rational expression into a polynomial plus a proper one using polynomial long division.
Is a constant over a polynomial a proper rational expression?
Yes. A nonzero constant like 5 is a polynomial of degree 0. If the denominator has degree 1 or higher, then 0 < deg(denominator), so the expression is proper. For example, 5/(x + 3) is proper.
Why do you need to know if a rational expression is proper?
Identifying whether a rational expression is proper is essential for partial fraction decomposition, a technique used in calculus for integration. Partial fractions can only be applied directly to proper rational expressions. If the expression is improper, you must first use polynomial long division to make it proper.

Proper Rational Expression vs. Improper Rational Expression

Proper Rational ExpressionImproper Rational Expression
DefinitionDegree of numerator < degree of denominatorDegree of numerator ≥ degree of denominator
Example(2x + 1) / (x² + x + 1)(x³ + 1) / (x² + x + 1)
Behavior as x → ∞Approaches 0Grows without bound (or approaches a constant if degrees are equal)
Partial fractionsCan be decomposed directlyMust first perform polynomial long division
Analogy to fractionsLike 3/7 (numerator < denominator)Like 7/3 (numerator ≥ denominator)

Why It Matters

You encounter proper rational expressions when learning partial fraction decomposition in precalculus and calculus. Recognizing whether a rational expression is proper tells you whether you can decompose it directly or need to perform long division first. This classification also matters in analyzing horizontal asymptotes: a proper rational expression always has a horizontal asymptote at y = 0, which is a quick graphing shortcut.

Common Mistakes

Mistake: Comparing the leading coefficients instead of the degrees of the polynomials.
Correction: The classification depends only on the highest power of x (the degree), not on the size of the coefficients. For instance, (100x)/(x²) is still proper because degree 1 < degree 2, even though 100 is much larger than 1.
Mistake: Thinking that equal degrees make a rational expression proper.
Correction: When the numerator and denominator have the same degree, the expression is improper, not proper. For example, (x² + 1)/(x² + 3) is improper because deg(numerator) = deg(denominator) = 2. The condition requires strictly less than.

Related Terms