Problem 1 - Precalculus Mathematics for Calculus

Question

For each rational function \(r,\) choose from (i)-(iv) the appropriate form for its partial fraction decomposition. $$r(x)=\frac{4}{x(x-2)^{2}}$$ $$\text { (i) } \frac{A}{x}+\frac{B}{x-2}$$ $$\text { (ii) } \frac{A}{x}+\frac{B}{(x-2)^{2}}$$ $$\text { (iii) } \frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^{2}}$$ $$\text { (iv) } \frac{A}{x}+\frac{B}{x-2}+\frac{C x+D}{(x-2)^{2}}$$

Step-by-Step Solution

Answer

The appropriate partial fraction decomposition form is (iii).

Step 1: Analyze the Denominator

The denominator of the given rational function is \(x(x-2)^2\). This indicates that it is composed of a linear factor \(x\) and a repeated linear factor \(x-2\) with multiplicity 2.

Step 2: Determine Decomposition Form

For a rational function with a linear factor \(x\) and a repeated linear factor \((x-2)\), we should include terms for each factor and each power of the repeated factor. This means using the form: \(\frac{A}{x} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\).

Step 3: Select the Appropriate Form

Looking at the options provided:- (i) does not account for the repeated factor's higher power- (ii) misses a term for the first power of \(x-2\)- (iii) matches exactly: \(\frac{A}{x} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\)- (iv) includes unnecessary terms for a polynomial numerator in the highest power, which is not requiredTherefore, the appropriate form for the partial fraction decomposition is (iii).

Key Concepts

Understanding Rational Functions

Rational functions are a foundational concept in algebra and calculus, where the function is expressed as the ratio of two polynomials. This means any rational function, say \( r(x) \), can be represented as \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials. The degree of \( q(x) \) in the denominator must always be greater than or equal to the degree of \( p(x) \) in the numerator for the function to be considered proper.
If the degree of \( p(x) \) is greater, polynomial long division may be used to rewrite the function before applying further analysis.
What makes rational functions interesting is their behavior around the values that make the denominator zero. These values lead to exceptions in the domain and play a critical role in understanding and applying partial fraction decomposition.

Exploring Linear Factors

At the heart of partial fraction decomposition are linear factors in polynomial denominators. A linear factor can be expressed as \( (x-a) \), where \( a \) is some constant. The denominator of a rational function may contain one or more of these linear factors.
The idea is to express the original complex fraction as a sum of simpler fractions, each having one of these linear factors in their denominator.

• A linear factor results in a partial fraction of form \( \frac{A}{x-a} \), for some constant \( A \).
• If the polynomial numerator is simpler than the linear factor, direct decomposition is possible.

This method allows easier integration and manipulation of complex rational functions, making the overall function simpler to handle.

The Role of Repeated Linear Factors

Repeated linear factors occur when a factor appears more than once in the denominator's polynomial. For example, a term \( (x-a)^n \) where \( n > 1 \) represents a repeated linear factor with \( a \) repeated \( n \) times.
Partial fraction decomposition must account for each occurrence of these factors individually to properly decompose the rational function.

• The decomposition reflects this by including separate terms for each power up to \( n \). For \( (x-a)^2 \), you will have terms like \( \frac{B}{x-a} + \frac{C}{(x-a)^2} \).
• This ensures that each power of the factor is properly represented, allowing us to solve coefficients for each term independently.

Handling repeated linear factors correctly ensures the accuracy of the decomposition and facilitates further analysis or integration of the function.

Understanding Polynomial Numerators

In partial fraction decomposition, polynomial numerators refer to the numerators in the decomposed terms, which are typically algebraic expressions in the form of constants or simple polynomials.
The form of the numerator often depends on the factors in the denominator:

• For simple linear factors \((x-a)\), the numerator is typically a constant \(A\).
• For repeated linear factors \((x-a)^n\), you may have numerators that are constants or even linear polynomials if necessary, such as \( B \) and \( C \) for increasing orders of the factor.
• In more complex cases, especially with quadratic factors, the numerator may involve multiple terms, though for rational functions with only linear factors, simpler numerators are typical.

Understanding how polynomials fit in the numerators ensures the correct setup for the decomposition process, preserving the balance of the originally given rational function.