Question

In Problems 5-16, determine whether the given rational expression is proper or improper. If the expression is improper, rewrite it as the sum of a polynomial and a proper rational expression.. $$ \frac{x}{x^{2}-1} $$

Short answer

The rational expression is proper.

Step-by-step solution

Define Proper and Improper Rational Expressions

A rational expression is proper if the degree of the numerator is less than the degree of the denominator. Otherwise, it is improper.

Determine the Degrees of Numerator and Denominator

The numerator is \(x\), which is a polynomial of degree 1. The denominator is \(x^2 - 1\), which is a polynomial of degree 2.

Classify the Given Rational Expression

Since the degree of the numerator (1) is less than the degree of the denominator (2), the rational expression \(\frac{x}{x^2 - 1}\) is proper.

Key concepts

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Rational expressions are fractions where both the numerator and the denominator are polynomials. Understanding them can be key in simplifying complex mathematical problems.

The rational expression in the exercise is \(\frac{x}{x^2 - 1}\). To determine if it's proper or improper, first recognize the definitions:

• A proper rational expression has the degree of the numerator less than the degree of the denominator.
• An improper rational expression has the degree of the numerator greater than or equal to the degree of the denominator.

For the given problem, the numerator is \(x\), a polynomial of degree 1, and the denominator is \(x^2 - 1\), a polynomial of degree 2. Since 1 is less than 2, this rational expression is proper.

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Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. To understand rational expressions, you must know about polynomials first.

In the exercise:

• The numerator \(x\) is a simple polynomial of degree 1.
• The denominator \(x^2 - 1\), factoring into \((x - 1)(x + 1)\), is a polynomial of degree 2.

Generally, polynomials are categorized by their degrees (highest power of the variable) and their terms (constants, variables with coefficients). Knowing these basics helps in identifying if a rational expression is proper or improper.

Polynomials can be classified by their degrees:

• Constant polynomial: Degree 0
• Linear polynomial: Degree 1
• Quadratic polynomial: Degree 2
• Cubic polynomial: Degree 3

In our case, recognizing the degrees of the polynomials involved lets us classify the rational expression correctly.

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The degree of a polynomial is a vital concept when working with rational expressions. The degree is the highest power of the variable in the polynomial.

For example, in the given exercise, look at the degrees:

• The numerator \(x\) has a degree of 1 (since \(x = x^1\)).
• The denominator \(x^2 - 1\) has a degree of 2 (the highest power of \(x\) is 2).

Properly identifying the degrees helps determine if a rational expression is proper or improper. Here, since 1 (numerator) < 2 (denominator), it confirms that \(\frac{x}{x^2 - 1}\) is proper.

It’s also useful when simplifying expressions, performing polynomial long division, or factoring. Degrees of polynomials are fundamental in understanding their behavior and how they interact in rational expressions.