Problem 2 - Calculus - AP Edition

Question

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{5}{x\left(x^{2}-4\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} \).

Step 1: Identify the Denominator

The first step in partial fraction decomposition is to identify the types of factors in the denominator of the given function. Here, the denominator is \(x(x^2-4)\). Notice that it consists of a linear factor \((x)\) and a quadratic factor \((x^2-4)\). The expression \(x^2-4\) can be further factorized as \((x-2)(x+2)\).

Step 2: Set Up Decomposition Structure

Based on the factorization, express the original rational function as a sum of fractions. Each linear factor will have a constant numerator, and each irreducible quadratic factor will have a linear numerator. Here, since \(x(x-2)(x+2)\) are all linear terms, assign single constants \(A\), \(B\), \(C\) to their numerators: \[ \frac{5}{x(x-2)(x+2)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} \]

Step 3: Write the Decomposition Without Solving

Combine these fractions back into one single fraction to ensure they add up to the original. Do this by obtaining a common denominator of \(x(x-2)(x+2)\). This step sets up the framework for eventually finding the values of \(A\), \(B\), and \(C\) if required: \[ \frac{5}{x(x^2-4)} = \frac{A(x-2)(x+2) + Bx(x+2) + Cx(x-2)}{x(x-2)(x+2)} \]. This confirms that our decomposition structure is in place. Notably, in this exercise, we are not asked to solve for \(A\), \(B\), and \(C\).

Key Concepts

Linear Factor

In mathematics, a linear factor refers to a component of a polynomial that is of degree one. It typically takes the form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is a variable. Linear factors are straightforward because their graph is a straight line.

• A linear factor in a polynomial indicates a single root or zero of that polynomial.
• Linear factors can be easily identified in polynomials of higher degrees, and they often play a crucial role in partial fraction decomposition.
• In the given function \(\frac{5}{x(x^2-4)}\), \(x\) is a linear factor.

Recognizing a linear factor helps in organizing rational functions into simpler parts, where each part can potentially be solved separately. This decomposition into linear factors, especially when paired with other higher degree factors, lays the groundwork for further analysis and calculation in mathematical problems.

Quadratic Factor

A quadratic factor in a polynomial is a component of degree two, typically expressed as \(ax^2 + bx + c\). Unlike linear factors, quadratic factors can have a more complex structure due to their squared component. They can represent parabolas when graphed.

• Quadratic factors might be reducible or irremediable depending on whether they can be further factored into linear factors.
• In the expression \(x(x^2-4)\), subtracting one quadratic factor is \(x^2-4\), which can further be decomposed into linear factors as \((x-2)(x+2)\).

Understanding quadratic factors, especially when they cannot be easily factored over the set of real numbers, is crucial in partial fraction decomposition. By converting them into product forms of linear factors when possible, one can simplify the analysis and calculations needed.

Factorization

Factorization involves breaking down a complex expression into simpler components, or factors, that when multiplied together yield the original expression. This process is fundamental across various fields of mathematics, including solving equations and simplifying algebraic expressions.

• The purpose of factorization is to simplify expressions or solve equations by finding their roots.
• In the partial fraction decomposition exercise, factorization is used to redefine the denominator \(x(x^2-4)\) as \(x(x-2)(x+2)\).
• Through factorization, the expression is simplified into its basic building blocks, making it easier to handle and solve in subsequent steps.

Understanding factorization strategies including recognizing common patterns, like difference of squares or perfect squares, enhances one’s ability to work through polynomials and find results effectively. In the context of partial fraction decomposition, it forms the foundation upon which the entire decomposition structure is built.