Problem 51 - Precalculus

Question

Determine whether each statement is true or false. The partial-fraction decomposition of a rational function \(\frac{f(x)}{(x-a)^{n}}\) has the form \(\frac{A_{1}}{(x-a)}+\frac{A_{2}}{(x-a)^{2}}+\cdots+\frac{A_{n}}{(x-a)^{n}}\) where all the numbers \(A_{i}\) are nonzero.

Step-by-Step Solution

Answer

False; not all coefficients \( A_i \) are necessarily nonzero.

Step 1: Understand Partial Fraction Decomposition

Partial fraction decomposition is a technique to express a rational function as a sum of simpler rational expressions. For a function \( \frac{f(x)}{(x-a)^n} \), its decomposition involves fractions where the denominators range from \( (x-a) \) to \( (x-a)^n \).

Step 2: Setup of Partial Fraction Decomposition

The decomposition of \( \frac{f(x)}{(x-a)^n} \) results in a sum of the form \( \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n} \). Each term in the decomposition has a distinct denominator, increasing in powers from \( 1 \) to \( n \).

Step 3: Analyze Coefficients

In partial fraction decomposition, the constants \( A_i \) are the coefficients that satisfy the equation when the decomposition is recombined. The values of \( A_i \) come from the solution of a system of linear equations. It is not secured by theorem that all \( A_i \) must be nonzero; some can be zero, depending on \( f(x) \).

Step 4: Determine the Truth Value of the Statement

The statement is false. While the form \( \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_n}{(x-a)^n} \) is correct for decomposing \( \frac{f(x)}{(x-a)^n} \), there is no requirement that each \( A_i \) is nonzero. It depends on the specific polynomial \( f(x) \) used.

Key Concepts

Rational Functions

Rational functions are expressions that represent the division of two polynomials. They take the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. These functions are widely used in mathematics because they can describe various real-world problems, such as rates of growth or decay, oscillations, and limits.
Working with rational functions often involves simplifying them or breaking them down into simpler components, which can make them easier to analyze and solve.

• Rational functions can have vertical asymptotes, which occur when the denominator equals zero, creating undefined values.
• Understanding how to manipulate and decompose rational functions is crucial in fields like calculus and algebra, where they're often applied to evaluate limits and integrals.

Linear Equations

Linear equations are foundational in algebra. They are equations of the first degree, meaning they involve only linear terms (no variables raised to a power higher than one). These equations have a simple form: \( ax + b = 0 \), where \( a \) and \( b \) are constants.
Linear equations are straightforward to solve because they have a consistent structure. When it comes to partial fraction decomposition, they play a pivotal role in determining the values of coefficients, like \( A_i \) in the decomposition.

• Solving a system of linear equations is often used to find the constants in partial fraction decomposition.
• The simplicity of linear equations allows for a quick resolution, making them an important tool for decomposing more complex rational expressions.

Polynomial

Polynomials are algebraic expressions that consist of variables and coefficients arranged in terms of whole number exponents. A typical polynomial has the structure: \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \), where the highest power of \( x \) determines the degree of the polynomial.
Polynomials are key to understanding rational functions and their decomposition. In a partial fraction decomposition, the numerator and the denominator are both polynomials.

• The degree of the numerator is critical when decomposing a rational function. It must be less than the degree of the denominator.
• Polynomials can be factored into simpler expressions, aiding the process of decomposition.
• Recognizing polynomial structures helps in setting up and solving the equations needed for finding partial fractions.

Denominators and Numerators

In rational functions, denominators and numerators are the two fundamental components that create the expression through division.

• The numerator is the top part of the fraction and determines whether the entire function can be simplified or requires decomposition.
• The denominator, being the bottom part, is especially crucial in partial fraction decomposition, as it determines the form of the simpler rational expressions obtained.

Understanding these parts is essential for transforming a complex rational function into simpler, more manageable pieces.
During decomposition, the polynomial in the denominator is typically factored, showcasing its roots and leading to the different terms used in decomposition.
Getting familiar with the roles of the numerator and the denominator aids in identifying proper forms for decomposition and ensuring that simplifications and operations maintain the function's integrity.