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. $$ \frac{x^{2}+5}{x^{2}-4} $$

Short answer

The rational expression \(\frac{x^{2}+5}{x^{2}-4}\) is improper and can be rewritten as \(1 + \frac{9}{x^{2}-4}\).

Step-by-step solution

Understand the form of the rational expression

A rational expression is proper if the degree of the numerator is less than the degree of the denominator. If the degree of the numerator is greater than or equal to the degree of the denominator, the rational expression is improper.

Determine the degrees of the numerator and the denominator

For the given rational expression \(\frac{x^{2}+5}{x^{2}-4}\), determine the degrees of the numerator and the denominator. The degree of the numerator \(x^{2}+5\) is 2, and the degree of the denominator \(x^{2}-4\) is also 2.

Classify the rational expression

Since the degree of the numerator (2) is equal to the degree of the denominator (2), the rational expression \(\frac{x^{2}+5}{x^{2}-4}\) is improper.

Perform polynomial division

To rewrite the improper rational expression as the sum of a polynomial and a proper rational expression, divide \(x^{2}+5\) by \(x^{2}-4\). Perform the division and find the quotient and the remainder.

Division calculation

Perform the division: \(\frac{x^{2}+5}{x^{2}-4} = 1 + \frac{9}{x^{2}-4} \). Here, the quotient is 1, and the remainder is 9, which we place over the original denominator \(x^{2}-4\).

Write the final result

The improper rational expression \(\frac{x^{2}+5}{x^{2}-4}\) can be rewritten as \(1 + \frac{9}{x^{2}-4}\), where \(1\) is the polynomial and \(\frac{9}{x^{2}-4}\) is the proper rational expression.

Key concepts

proper rational expression

A rational expression is a fraction where both the numerator and the denominator are polynomials. We call it a 'proper rational expression' when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. For example, consider \(\frac{x}{x^2 + 1}\). Here, the numerator is of degree 1 and the denominator is of degree 2. Since the numerator's degree (1) is less than the denominator's degree (2), this is a proper rational expression.

improper rational expression

An 'improper rational expression' is when the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. For instance, take \(\frac{x^3 + 3x + 2}{x^2 - x}\). The numerator is of degree 3 and the denominator is of degree 2, making this an improper rational expression because the numerator's degree (3) is greater than the denominator's degree (2). In such cases, we often use polynomial division to rewrite the expression as a sum of a polynomial and a proper rational expression.

polynomial division

Polynomial division is much like regular long division but for polynomials. It's used to simplify improper rational expressions. Let's go through a quick example: Suppose we want to divide \(\frac{x^2 + 5}{x^2 - 4}\). Our steps would be:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire denominator by this result and subtract from the numerator.
• Continue this process with the resulting polynomial until the remaining polynomial has a degree less than the denominator.

In our example, we find the quotient is 1 and the remainder is 9, so \(\frac{x^2 + 5}{x^2 - 4} = 1 + \frac{9}{x^2 - 4}\).

degree of polynomials

The 'degree of a polynomial' is the highest power of the variable in the polynomial. For example, in the polynomial \(x^4 + 3x^2 + 2\), the degree is 4, because the highest power of x is 4. When comparing degrees for rational expressions and polynomial division, understand which terms are influencing the polynomial's behavior. In the expression \(\frac{x^{2}+5}{x^{2}-4}\), both numerator and denominator have a degree of 2, making it an improper rational expression. Recognizing these degrees helps you determine the appropriate method for simplifying the expression.