Problem 12 - Precalculus with Limits

Question

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{2 x-3}{x^{3}+10 x}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \(\frac{A}{x} + \frac{Bx+C}{x^2+10}\)

Step 1: Simplify the denominator

The denominator \(x^{3}+10 x\) can be simplified by taking out a common factor of x. Hence, this gives \(x(x^2+10)\).

Step 2: Set up the form of the partial fraction decomposition

We need to set up a fraction for each factor in the denominator. The fraction for \(x\) will look like \(\frac{A}{x}\), and the fraction for \(x^2+10\) will look like \(\frac{Bx+C}{x^2+10}\). Bx is used in the numerator as \(x^2 + 10\) is a quadratic with degree 2.

Step 3: Combine the partial fractions

The combined form of these fractions is \(\frac{A}{x} + \frac{Bx+C}{x^2+10}\). This is the form of the partial fraction decomposition of the given rational expression, indicating what constants would need to be solved for if required.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions are fundamental in algebra as they appear frequently in calculus and beyond.

Consider the rational expression \( \frac{2x-3}{x^3+10x} \). Here, \( 2x-3 \) is the numerator, and \( x^3+10x \) is the denominator. To simplify, analyze, or solve these expressions, we often decompose them into simpler parts, especially when integrating or differentiating in calculus. This process is known as partial fraction decomposition, a technique aimed at breaking down complex rational expressions into simpler pieces that are easier to work with.

When dealing with rational expressions, it's important to ensure that the degree of the numerator is less than the degree of the denominator; otherwise, the expression is improper and needs to be simplified first. If a rational expression is improper, polynomial division might be necessary before proceeding with other techniques.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler polynomial multipliers. This step is essential during the partial fraction decomposition as it reveals the roots and simplifies the expression.

In the given rational expression \( \frac{2x-3}{x^3+10x} \), the denominator \( x^3 + 10x \) can be tricky to work with in its original form. Thus, we start by factoring it. By identifying and factoring out the greatest common factor (GCF), we simplify this expression:

• Notice that both terms, \( x^3 \) and \( 10x \), have a common factor of \( x \).
• Extracting \( x \), leaves us with \( x(x^2 + 10) \).

This factoring is crucial as it allows us to separate the rational expression into simpler fractions, paving the way for partial fraction decomposition.

Factoring polynomials is not only integral for simplifying expressions; it is also a foundational skill in solving equations, finding roots, and performing integration.

Simplifying Denominators

Simplifying denominators is a key step in the partial fraction decomposition as it allows for setting up the subsequent expressions correctly. Once a polynomial is factored, writing the corresponding partial fractions is straightforward.

From the factored form \( x(x^2 + 10) \), we assign a single fraction to each factor:

• The linear factor \( x \) yields a fraction of the form \( \frac{A}{x} \).
• The quadratic factor \( x^2 + 10 \) generates a fraction of the form \( \frac{Bx+C}{x^2+10} \), where \( Bx + C \) represents a polynomial of one degree less than \( x^2+10 \).

By simplifying the denominator this way, setting up the partial fractions becomes a matter of aligning terms appropriately. This preparation step is crucial, especially when it comes to solving for the constants \( A \), \( B \), and \( C \), as it forms the basis for understanding how the partial fractions are structured. Knowing how to decompose and simplify these components aids significantly in understanding complex algebraic manipulations, as well as in practical computations such as integration.