Problem 5 - 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. \(\frac{x^{2}}{(x-3)\left(x^{2}+4\right)}\)

Step-by-Step Solution

Answer

\(\frac{A}{x-3} + \frac{Bx+C}{x^2+4}\).

Step 1: Identifying the Denominator

The denominator of the given fraction is \((x-3)(x^2+4)\). Here, \(x-3\) is a linear factor, and \(x^2+4\) is an irreducible quadratic factor.

Step 2: Setting Up the Partial Fraction Form

For linear factors like \(x-3\), we write a term \(\frac{A}{x-3}\). For quadratic factors like \(x^2+4\), we write a term \(\frac{Bx + C}{x^2 + 4}\). Thus, the partial fraction decomposition is set up as follows: \[\frac{x^2}{(x-3)(x^2+4)} = \frac{A}{x-3} + \frac{Bx+C}{x^2+4}\]

Key Concepts

Linear Factor

In partial fraction decomposition, a **linear factor** is a factor in the denominator that is a polynomial of degree one. For example, the expression \(x - 3\) is a linear factor. Working with linear factors is straightforward, due to their simple form.
Here's what you usually do with a linear factor in partial fraction decomposition:

• For each linear factor like \(x-a\), you'll have a term such as \(\frac{A}{x-a}\).
• \(A\) represents a constant that will be determined later when the entire decomposition is simplified and solved for specific values.

Recognizing linear factors is the first step towards setting up the partial fraction decomposition correctly. This allows you to express the original fraction in terms of simpler terms. By breaking down the fraction into these simpler components, each piece can be understood and managed more easily.

Irreducible Quadratic Factor

An **irreducible quadratic factor** is a quadratic polynomial that cannot be factored further into real linear terms. An example from our function is \(x^2 + 4\). In the context of partial fraction decomposition, dealing with irreducible quadratic factors takes an additional step compared to linear factors.
Here's how you handle such factors:

• For each irreducible quadratic factor like \(x^2 + b\), the associated term in the partial fraction form will look like \(\frac{Bx + C}{x^2 + b}\).
• The letters \(B\) and \(C\) represent unknown constants that will be calculated later.

These terms incorporate \(x\) in the numerator to account for the extra degree found in the quadratic factor. Through this method, you can start solving the equation by matching coefficients, once the expression is set up in partial fraction form.

Partial Fraction Form

The **partial fraction form** is an expression that represents a more complex rational function as a sum of simpler fractions. These simpler components are easier to integrate or manipulate mathematically. In our given exercise, the original fraction is \(\frac{x^2}{(x-3)(x^2+4)}\).
The right partial fraction form for this example is:

• \(\frac{A}{x-3}\) related to the linear factor \(x-3\).
• \(\frac{Bx + C}{x^2 + 4}\) related to the irreducible quadratic factor \(x^2 + 4\).

The partial fraction decomposition makes it easier to analyze the function because each fraction now matches the form based on the original denominator's factors.
To further solve for \(A, B,\) and \(C\), you would need to multiply through by the original denominator and equate coefficients from both sides of the equation. This systematic approach allows for more straightforward integration and other operations in calculus.