Problem 55 - College Algebra

Question

Find the partial fraction expansion. \(\frac{x^{2}+4}{(x+1)^{3}}\)

Step-by-Step Solution

Answer

The partial fraction expansion is \( \frac{1}{x+1} - \frac{2}{(x+1)^2} + \frac{5}{(x+1)^3} \).

Step 1: Understand the Problem

We need to express the given rational function \( \frac{x^2 + 4}{(x+1)^3} \) as a sum of simpler fractions whose denominators are powers of \( (x+1) \). This is called partial fraction decomposition and will simplify the integration or simplification of the expression.

Step 2: Set Up the Partial Fraction Structure

Since the denominator is \((x+1)^3\), the partial fraction decomposition should include terms with denominators \(x+1\), \((x+1)^2\), and \((x+1)^3\). Therefore, we express:\[ \frac{x^2 + 4}{(x+1)^3} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \]where \( A \), \( B \), and \( C \) are constants to be determined.

Step 3: Combine the Fractions

Combine the right hand side into a single fraction:\[ \frac{A(x+1)^2 + B(x+1) + C}{(x+1)^3} \]Set this equal to the original expression:\[ \frac{A(x+1)^2 + B(x+1) + C}{(x+1)^3} = \frac{x^2 + 4}{(x+1)^3} \]

Step 4: Clear the Denominators

Multiply both sides by \((x+1)^3\) to clear the denominators, obtaining:\[ A(x+1)^2 + B(x+1) + C = x^2 + 4 \]

Step 5: Expand and Simplify

Expand the equation on the left side:\[ A(x^2 + 2x + 1) + B(x+1) + C = x^2 + 4 \]\[ Ax^2 + 2Ax + A + Bx + B + C = x^2 + 4 \]

Step 6: Equate Coefficients

Equate the coefficients of corresponding terms on both sides:- For \( x^2 \): \( A = 1 \)- For \( x \): \( 2A + B = 0 \)- For the constant term: \( A + B + C = 4 \)

Step 7: Solve the System of Equations

First, solve \( A = 1 \) to find \( A \).Substitute \( A = 1 \) into the equations:- From \( 2A + B = 0 \), we have \( 2(1) + B = 0 \) giving \( B = -2 \).- Substitute \( A = 1 \) and \( B = -2 \) in \( A + B + C = 4 \): \[ 1 - 2 + C = 4 \] \( C = 5 \).

Step 8: Write the Decomposition

The partial fraction decomposition is:\[ \frac{x^2 + 4}{(x+1)^3} = \frac{1}{x+1} - \frac{2}{(x+1)^2} + \frac{5}{(x+1)^3} \]

Step 9: Verify the Solution

Expand and simplify the right-hand side to ensure it matches the original expression when combined. Calculate any substitutions if needed and confirm the solution correctness.

Key Concepts

Rational Function

A rational function is a fraction where both the numerator and the denominator are polynomials. In the problem given, we have a rational function represented by \(\frac{x^2+4}{(x+1)^3}\). This function is called 'rational' because it makes use of ratios; specifically, the ratio of the polynomial \(x^2+4\) to another polynomial \((x+1)^3\).
Understanding rational functions is crucial for operations such as calculus and algebra because these functions frequently arise in problems involving rates, measurements, or any scenario requiring division.

• The simplest rational functions are those where both the numerator and the denominator are linear polynomials, like \(\frac{ax+b}{cx+d}\).
• More complex cases appear when higher degree polynomials or multiplicative terms like \((x+1)^n\) exist in the denominator.

The essence of partial fraction decomposition, as seen in the solution, helps to break down complicated rational functions into simpler fractions that are easier to work with, especially for further calculations in calculus.This decomposition results in a set of fractions each with simpler, often linear, denominators. It makes integration or differentiation tasks more manageable.

Integration

Integration is a fundamental concept in calculus used to find the area under a curve or the total accumulation of quantities. When dealing with rational functions, integration can be tricky due to the complexity of the expressions.
Partial fraction decomposition, as illustrated in the solution, simplifies the integration of a rational function. By breaking the original function into simpler components, each part can be integrated separately. This results in a solution that is far easier to compute and interpret.
Here is why partial fraction decomposition facilitates integration:

• Each decomposed fraction usually involves simpler algebraic terms.
• Standard integration techniques, such as direct integration formulas, are applicable to these simpler terms.

For example, the decomposition, \(\frac{1}{x+1} - \frac{2}{(x+1)^2} + \frac{5}{(x+1)^3}\), means each term can be integrated individually using basic integration rules.
This systematic approach takes what could be a complex problem and divides it into manageable parts, simplifying the entire integration process.

Denominator

In rational functions, the denominator is the lower part of a fraction that indicates the number of equal parts the numerator is divided by. In our given problem, the denominator is \((x+1)^3\). This specific form suggests repeated linear factors.
Understanding denominators is crucial because they determine the nature of the fraction and the possible simplifications or transformations, such as partial fraction decomposition. They also influence the overall behavior of the function.
For decomposing any rational function, one must consider:

• The degree of the polynomial, which affects the terms needed in the partial fractions.
• Repeated factors, like \((x+1)^3\), that lead to terms involving polynomial powers in the decomposition structure.

Breaking down the denominator simplifies the expression, facilitating processes like integration. This understanding allows us to express a complex rational function as a sum of simpler components, each with its own factor in the denominator.
In practical terms, this translates complex integrals or functions into easier-to-handle components, simplifying calculation and application.

System of Equations

A system of equations involves multiple equations that are solved collectively because they share variables. In the context of partial fraction decomposition, a system of equations helps to find the unknown constants that satisfy the decomposition.
In the given solution, we solved the system:

• \(A = 1\)
• \(2A + B = 0\)
• \(A + B + C = 4\)

This system arises when equating coefficients of the decomposed parts to match the original polynomial. Solving these equations tells us the specific values of \(A\), \(B\), and \(C\) that make the decomposition valid.
Systems of equations can be tackled using various methods like substitution, elimination, or matrices. In our problem, a simple substitution method provided the solution:
Identifying \(A = 1\), led us to find \(B\) and \(C\) through straightforward substitution. These techniques make handling complex expression breakdown straightforward, efficient, and clear, especially when confirmed by re-combining the terms to match the original function.