Problem 9 - Algebra and Trigonometry

Question

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{x^{3}+x+1}{x(2 x-5)^{3}\left(x^{2}+2 x+5\right)^{2}}\)

Step-by-Step Solution

Answer

\(\frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex + F}{x^2 + 2x + 5} + \frac{Gx + H}{(x^2 + 2x + 5)^2}\).

Step 1: Analyze the Denominator

Identify the different types of factors in the denominator: we have the linear factor \(x\), the repeated linear factor \((2x-5)^3\), and the repeated irreducible quadratic factor \((x^2 + 2x + 5)^2\).

Step 2: Set Up Partial Fractions for Linear Factors

For the linear factor \(x\), assign a term \(\frac{A}{x}\) where \(A\) is a constant to be determined.

Step 3: Set Up Partial Fractions for Repeated Linear Factor

For the repeated linear factor \((2x-5)^3\), assign terms for each power up to 3: \(\frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3}\).

Step 4: Set Up Partial Fractions for Repeated Irreducible Quadratic Factor

For \((x^2 + 2x + 5)^2\), assign terms in the form of \(\frac{Ex + F}{x^2 + 2x + 5}\) and \(\frac{Gx + H}{(x^2 + 2x + 5)^2}\).

Step 5: Combine All Parts

Combine all terms together to form the full expression for the partial fraction decomposition: \(\frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex + F}{x^2 + 2x + 5} + \frac{Gx + H}{(x^2 + 2x + 5)^2}\).

Key Concepts

Linear Factors

In the world of algebra and calculus, linear factors are a core concept of partial fraction decomposition. A linear factor is a polynomial of the first degree, meaning it can be expressed in the form of \(ax + b\), where \(a\) and \(b\) are constants, and \(a\) is not zero. In the given expression, the factor \(x\) is an example of a simple linear factor. When decomposing a rational function, each linear factor receives a fractional term consisting of a constant numerator and the linear factor itself as the denominator. For instance, for \(x\), we assign a term \(\frac{A}{x}\).

• This approach helps in simplifying complex rational expressions.
• It makes integration and other calculus operations more manageable.

Understanding linear factors and their role in decomposition is crucial for tackling more advanced algebraic problems.

Repeated Linear Factors

Repeated linear factors emerge in partial fraction decomposition when a linear factor is raised to a power greater than one. In our expression, \( (2x-5)^3 \) exemplifies this concept, where the linear factor \((2x-5)\) is repeated three times.To decompose such factors, we write separate terms for each power starting from one up to the factor's exponent. For \((2x-5)^3\), we assign:

• \(\frac{B}{2x-5}\)
• \(\frac{C}{(2x-5)^2}\)
• \(\frac{D}{(2x-5)^3}\)

This ensures each degree of the factor is accounted for, allowing us to adequately cover all potential influences of this term on the function's behavior. Recognizing and utilizing repeated linear factors is key to accurate partial fraction decomposition.

Irreducible Quadratic Factors

Irreducible quadratic factors are polynomial expressions of degree two that do not have real roots. In other words, they can't be factored further into real linear factors. The term \((x^2 + 2x + 5)\) in our denominator is an example of an irreducible quadratic factor.For such factors in partial fraction decomposition, we use a numerator of the form \(Ax + B\) rather than a single constant. This accounts for any polynomial expression up to the degree of the quadratic factor. For example:

• \(\frac{Ex + F}{x^2 + 2x + 5}\)
• \(\frac{Gx + H}{(x^2 + 2x + 5)^2}\)

The extra complexity in the numerator provides the flexibility needed to capture the subtler behaviors introduced by quadratic terms. Mastering this concept is important for advanced calculus and differential equations.

Rational Functions

Rational functions are fractions in which both the numerator and the denominator are polynomials. The study and breakdown of rational functions are central to understanding partial fraction decomposition, as seen in our expression \(\frac{x^{3}+x+1}{x(2 x-5)^{3}(x^{2}+2 x+5)^{2}}\).The goal of partial fraction decomposition is to simplify a complex rational function into simpler, easily manageable parts. This is achieved by breaking down the rational function based on its denominator factors, whether they are linear, repeated, or irreducible quadratic factors.

• This simplification aids in integrating rational functions, where direct integration might be difficult.
• It is also useful in solving differential equations.

By dissecting a rational function into its basic components, we gain deeper insights and an enhanced ability to solve mathematical problems effectively.