Problem 1 - Calculus of a Single Variable

Question

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{4}{x^{2}-8 x} $$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational function is \(\frac{A}{x}+\frac{B}{x-8}\).

Step 1: Factorize the denominator

In our problem, the denominator is \(x^{2}-8x\). If we factor out the common term 'x' we obtain \(x(x-8)\).

Step 2: Write the form of partial fraction decomposition

Based on the factorization, we can now perform the partial fraction decomposition. The general form is \(\frac{A}{x}+\frac{B}{x-8}\) for some constants A and B.

Key Concepts

Rational Expressions

Rational expressions are quite similar to fractions, but instead of numbers, they involve polynomials. A rational expression is a ratio of two polynomials, essentially like a fraction where the numerator and denominator are polynomial functions. For example, in the expression \( \frac{4}{x^2 - 8x} \), we see that the numerator is the constant 4, while the denominator is a polynomial \( x^2 - 8x \). Understanding how to manipulate these expressions through operations like addition, subtraction, and decomposition into partial fractions is key in solving complex algebraic equations.

These expressions can appear in many mathematical contexts, requiring techniques for simplification or breakdown. By reducing them into simpler components, such as partial fractions, their integrals and sums can be easier to handle, especially in calculus applications.

Factorization

Factorization is a crucial step when working with rational expressions, particularly in partial fraction decomposition. It involves breaking down a polynomial into simpler multiplicative components.

In our example, the denominator \( x^2 - 8x \) can be factorized by identifying common factors. By taking 'x' as a common factor, it can be rewritten as \( x(x - 8) \). This process is pivotal as it lays the groundwork for partial fraction decomposition by creating simpler expressions that add up to the original rational expression.

Effective factorization not only simplifies calculations but also reveals insights into the behavior and characteristics of the polynomial. It is an essential skill that aids in solving complex algebraic problems efficiently.

Denominator

The denominator in rational expressions plays a critical role as it determines the form and possibilities of the partial fraction decomposition. It is the polynomial found at the bottom of the fraction — the part of the rational expression that divides the numerator.

For the expression \( \frac{4}{x^2 - 8x} \), the denominator is \( x^2 - 8x \), which can be rewritten by factorization as \( x(x - 8) \).

When performing partial fraction decomposition, understanding the denominator is important because its roots and factored form help in determining the terms of the decomposition. Each factor in the denominator with a linear term, as in our example, corresponds to a term in the partial fraction where constants will eventually be determined.

Constants

In partial fraction decomposition, constants play an integral role in expressing the rational expression as a sum of simpler fractions. These constants represent unknown numbers that will be calculated after setting up the decomposition form.

In our example, after factorizing the denominator into \( x(x - 8) \), the partial fraction form becomes \( \frac{A}{x} + \frac{B}{x - 8} \). Here, \( A \) and \( B \) are constants that need to be found to satisfy the equality with the original expression.

While solving these constants isn't part of the initial setup, recognizing their presence is essential. They serve to balance the equation, ensuring that when recombined, the sum of these fractions accurately represents the original rational expression. Understanding and solving for these constants helps students develop a deeper grasp of how complex expressions can be broken down and manipulated.