Problem 2 - Precalculus with Limits

Question

If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ______.

Step-by-Step Solution

Answer

The fraction is called 'improper'.

Step 1: Understand the Term - Rational Expression

A rational expression is a fraction where both numerator and denominator are polynomials. For example, \( \frac{x^2 + 1}{x - 2} \) is a rational expression.

Step 2: Understand the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial \(2x^3 - x^2 + 1\), 3 is the degree since it is the highest power of the variable x.

Step 3: Classification of Rational Expressions

When the degree of the numerator is greater than or equal to the degree of the denominator in a rational expression, the fraction is called 'improper'. If the degree of the numerator is less than the degree of the denominator, the fraction is termed 'proper'.

Key Concepts

Improper Fractions

Improper fractions in the context of rational expressions tell us about the relationship between the numerator and the denominator. Simply put:

• When the degree of the top part (numerator) is greater than or equal to the degree of the bottom part (denominator), the fraction is improper.

Why does this matter? Because it affects how we simplify or analyze these fractions. For instance, if you have the rational expression \(\frac{x^3 + x^2 + 5}{x^2 + 3}\), the degree of the numerator (3) is greater than that of the denominator (2). As a result, this is an improper fraction.
When working with improper fractions, one useful strategy is to perform polynomial long division. This allows for a simpler mixed expression, often making it easier to work with. By dividing the numerator by the denominator, you can isolate any whole number component from the rational part of the expression.

Polynomial Degree

Understanding the degree of a polynomial is crucial when working with rational expressions. The degree is the highest power of the variable that appears in the polynomial. For example, consider the polynomial \(3x^5 + 2x^3 - x + 7\). The highest exponent of \(x\) here is 5, so this polynomial has a degree of 5.
Why is degree important? The degree plays a crucial role in categorizing rational expressions as proper or improper. By comparing the degrees of the numerator and denominator:

• If the numerator's degree is higher or equal, you have an improper fraction.
• If it is lower, then it's a proper fraction.

Knowing the polynomial degree helps when simplifying expressions, solving equations, or determining the behavior of functions at extreme values of \(x\). Thus, always pay close attention to these exponents when analyzing rational expressions.

Proper Fractions

A proper fraction in the realm of rational expressions is much like the proper fractions you may know from simple arithmetic. Here, however, we are dealing with polynomials instead of plain numbers.

• For a fraction to be considered proper, the degree of the numerator must be less than the degree of the denominator.

Consider \(\frac{x^2 + 3}{x^3 - x}\) as an example. The numerator has a degree of 2, while the denominator has a degree of 3. Because the numerator’s degree is less, this is a proper fraction.
The significance of proper fractions lies in their simplicity. These fractions do not need polynomial division for simplification because the numerator is 'contained' under the degree of the denominator. Still, they can often be further simplified by factoring or canceling terms, especially when solving equations or evaluating limits.