Question

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. $$ \frac{3 x^{2}-2}{x^{2}-1} $$

Short answer

The expression is improper. It can be rewritten as: \(3 + \frac{1}{x^{2}-1}\).

Step-by-step solution

- Identify Degrees of Polynomial in Numerator and Denominator

Determine the degrees of the polynomials in the numerator and the denominator. The degree of the numerator, \(3x^2 - 2\), is 2. The degree of the denominator, \(x^2 - 1\), is also 2.

- Determine if the Expression is Proper or Improper

A rational expression is proper if the degree of the numerator is less than the degree of the denominator. Since the degrees are equal, the rational expression \( \frac{3x^2 - 2}{x^2 - 1} \) is improper.

- Perform Polynomial Division

Perform polynomial long division to divide \(3x^2 - 2\) by \(x^2 - 1\). The first term of the quotient is 3, as \(3x^2\) divided by \(x^2\) gives 3. Multiply the denominator by 3: \(3(x^2 - 1) = 3x^2 - 3\). Subtract this from the original numerator: \[(3x^2 - 2) - (3x^2 - 3) = 1.\]Therefore, the quotient is 3 and the remainder is 1.

- Express the Rational Expression as a Sum

Rewriting the rational expression using the quotient and the remainder, we have:\[ \frac{3x^2 - 2}{x^2 - 1} = 3 + \frac{1}{x^2 - 1}.\]Thus, the expression is written as the sum of a polynomial (3) and a proper rational expression \( \frac{1}{x^2 - 1} \).

Key concepts

Polynomial Division

Polynomial division is similar to the long division of numbers. It involves breaking down a complex polynomial into simpler parts. The goal is to divide the numerator polynomial by the denominator polynomial to find the quotient and the remainder. This process helps simplify improper rational expressions.

For example, let's divide \(3x^2 - 2\) by \(x^2 - 1\). Start by dividing the leading terms: \(3x^2\) divided by \(x^2\) gives you \(3\). Multiply \(3\) by the entire denominator \(x^2 - 1\) to get \(3x^2 - 3\). Then, subtract this result from the original numerator to obtain the remainder. The final step is to express the original rational expression as the quotient plus the fraction formed by the remainder over the original denominator.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable in the polynomial. It plays a crucial role in determining the nature of rational expressions.

For instance:

• The polynomial \(3x^2 - 2\) has a degree of 2 because the highest exponent of \(x\) is 2.
• The polynomial \(x^2 - 1\) also has a degree of 2 for the same reason.

When comparing degrees, if the degree of the numerator is greater than or equal to the degree of the denominator, the rational expression is considered improper. Otherwise, it is proper. In our case, the degrees are equal, making the expression improper.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They can be either proper or improper.

A proper rational expression has a numerator with a degree less than the denominator’s degree. Conversely, an improper rational expression has a numerator with a degree greater than or equal to the denominator’s degree. For example, \(\frac{3x^2 - 2}{x^2 - 1}\) is improper, as the numerator and denominator both have degrees of 2.

To work with improper rational expressions, we often use polynomial division to rewrite them as a sum of a polynomial and a proper rational expression, providing a simpler form that is easier to analyze and work with.

Algebraic Fractions

Algebraic fractions are expressions where both the numerator and the denominator are polynomials. Understanding how to manipulate these fractions is essential in algebra.

When dealing with improper algebraic fractions, you need to split them into simpler forms using polynomial division. This process involves:

• Dividing the numerator by the denominator to find the quotient.
• Multiplying the quotient by the denominator and subtracting the result from the original numerator to find the remainder.
• Expressing the original fraction as the sum of the quotient and the fraction made up of the remainder and the denominator.

This method makes complex algebraic expressions simpler to solve or integrate, as seen in the breakdown of \(\frac{3x^2 - 2}{x^2 - 1}\) into \(3 + \frac{1}{x^2 - 1}\).