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. $$ \frac{x(x-1)}{(x+4)(x-3)} $$

Short answer

The expression is improper. It can be rewritten as \(1 + \frac{-13x + 12}{x^2 + x - 12}\)

Step-by-step solution

- Define Proper and Improper Rational Functions

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

- Determine the Degree of the Numerator and Denominator

Identify and compare the degrees of the numerator and the denominator. The numerator is \(x(x-1)\), which expands to \(x^2 - x\). The degree of the numerator is 2. The denominator is \((x+4)(x-3)\), which expands to \(x^2 + x - 12\). The degree of the denominator is also 2.

- Identify the Rational Expression as Proper or Improper

Since the degrees of the numerator and the denominator are both 2, the rational expression \(\frac{x(x-1)}{(x+4)(x-3)}\) is improper.

- Rewrite the Improper Rational Expression

To rewrite the improper rational expression as the sum of a polynomial and a proper rational expression, perform polynomial long division. Divide \(x^2 - x\) by \(x^2 + x - 12\). The division gives 1 with a remainder of − \(13x - 12\). Therefore, the rational expression can be rewritten as: \[\frac{x(x-1)}{(x+4)(x-3)} = 1 + \frac{-13x + 12}{x^2 + x - 12} \]

- Simplify the Results

The polynomial part is 1, and the proper rational expression part is \(\frac{-13x + 12}{x^2 + x - 12}\)

Key concepts

Proper vs Improper Rational Functions

In mathematics, it's essential to distinguish between proper and improper rational functions. A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding whether it is proper or improper helps us decide how to simplify or manipulate it.

A proper rational function has a numerator with a degree less than the degree of its denominator. Conversely, an improper rational function has a numerator with a degree that is greater than or equal to the degree of its denominator.

For instance, if we have the rational expression \(\frac{2x^2 + 3}{x^3 + 5x + 1}\), this is proper because the degree of the numerator (2) is less than the degree of the denominator (3). On the other hand, \(\frac{4x^4 + 5x^2 + 1}{x^2 + 2}\) is improper since the degree of the numerator (4) is higher than the degree of the denominator (2).

In our given example, with the rational function \(\frac{x(x-1)}{(x+4)(x-3)}\), expanding the numerator gives us \(x^2 - x\) and the denominator \(x^2 + x - 12\). Both have a degree of 2, making the rational function improper.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials similar to the long division of numbers. This technique allows us to rewrite improper rational expressions into a more manageable form: the sum of a polynomial and a proper rational expression. Let's break down how it's done.

First, let's consider dividing \(x^2 - x\) by \(x^2 + x - 12\). You start by looking at the leading terms of both the numerator and the denominator. Here, both leading terms are \(x^2\).

1. Divide \(x^2\) by \(x^2\) to get 1.
2. Multiply the entire denominator by this quotient: \(1 \times (x^2 + x - 12) = x^2 + x - 12\).
3. Subtract this result from the numerator: \(x^2 - x - (x^2 + x - 12) = -2x + 12\).
4. Since the remaining polynomial (\(-2x + 12\)) has a degree less than that of the denominator, we stop here.

This shows us that \(\frac{x^2 - x}{x^2 + x - 12}\) can be written as 1 plus a proper rational expression, specifically \(\frac{-2x + 12}{x^2 + x - 12}\).

Rewriting Rational Expressions

Once we have identified an improper rational function, we can make it simpler by rewriting it as a sum of a polynomial and a proper rational expression. This makes further operations, integrations, or solving equations more manageable.

Using our ongoing example of \(\frac{x(x-1)}{(x+4)(x-3)}\), through polynomial long division, we found that:

\(\frac{x(x-1)}{(x+4)(x-3)} = 1 + \frac{-13x + 12}{x^2 + x - 12}\).

This transformation is useful because now the rational expression \(\frac{-13x + 12}{x^2 + x - 12}\) is proper. We have isolated the whole number (1) and simplified what was initially a seemingly complex fraction.

Rewriting in this manner also prepares the expression for integration or other advanced calculus techniques since working with simpler forms is generally more straightforward. This strategy applies to various contexts where handling polynomials and rational expressions is required, highlighting the importance of mastering polynomial long division and proper vs. improper rational expressions.