Problem 15 - 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. $$\frac{x-1}{x\left(x^{2}+1\right)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition is \(\frac{x-1}{x(x^2+1)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2}\).

Step 1: Identify the denominator structure

The denominator of the rational expression is \(x(x^2+1)^2\). It can be seen as the product of two functions: \(x\) and \((x^2+1)^2\). Both of these needs to be accounted for in the partial fraction decomposition.

Step 2: Set up the decomposition

Decompose into partial fractions. Each factor of the denominator defines a term of the decomposition. Also consider the repeated roots associated with \((x^2+1)^2\).

Step 3: Write the form of the decomposition

The form of the partial fraction decomposition is as follows: \(\frac{x-1}{x(x^2+1)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2}\). Here A, B, C, D, and E represent coefficients which we are not asked to solve for in this exercise.

Key Concepts

Rational Expression

A rational expression is quite similar to a regular fraction. It consists of a numerator and a denominator, but instead of only integers, these are often polynomials. In simpler terms, a rational expression is the fraction of one polynomial over another. For example, \( \frac{x-1}{x(x^2+1)^2} \) is a rational expression.We often work with rational expressions in algebra to simplify complex equations, solve for variables, or integrate functions. These expressions play a crucial role in calculus and advanced algebra, as they align closely with real-world relationships and data modeling. Rational expressions are a big stepping stone towards understanding more complex mathematical topics.

Denominator Structure

The denominator structure is key when working with rational expressions, especially during partial fraction decomposition. Analyzing the denominator helps us determine how to decompose the rational expression into its simplest components. In our example, the denominator is \( x(x^2+1)^2 \).- **Factor Breakdown**: - The denominator can be viewed as a product of two separate parts: \( x \) and \( (x^2 + 1)^2 \). - Each of these factors will dictate the format of our decomposition terms.Understanding the structure means recognizing these factors and seeing if they are linear, irreducible quadratic, or higher powers. This recognition informs how you set up your partial fractions, ensuring all deterrent terms are accounted for in the decomposition.

Repeated Roots

In decomposing rational expressions, repeated roots refer to a factor in the denominator that appears more than once. They require special attention for accurate decomposition. In our example, the \( (x^2 + 1)^2 \) is a repeated root. This means \((x^2 + 1)\) is squared, and it asks for more intricate decomposition terms.To handle repeated roots: - Include a separate term for each power. Start from the first power up to the highest. - In this exercise, that means creating terms for \((x^2+1)\) and \((x^2+1)^2\).- This ensures every aspect of the root's behavior is captured, making the decomposition accurate.

Decomposition Terms

The crux of partial fraction decomposition lies in accurately setting up the decomposition terms. Given the denominator structure and repeated roots, determine what terms are necessary. In the expression \( \frac{x-1}{x(x^2+1)^2} \), we decompose it to:- \( \frac{A}{x} \) for the linear \( x \).- \( \frac{Bx+C}{x^2+1} \) for the first power of the quadratic \( x^2+1 \).- \( \frac{Dx+E}{(x^2+1)^2} \) for the repeated root \((x^2+1)^2\).Each of these terms includes coefficients \( A, B, C, D, \) and \( E \), highlighting how the expression can be split. They become placeholders, ready to be solved or simplified, according to the needs of the particular algebraic operation you're performing. This makes partial fraction decomposition powerful in simplifying and integrating complex rational expressions.