Problem 55 - College Algebra

Question

For the following exercises, 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: Identify the Type of Partial Fractions

The denominator \((x+1)^3\) is a single repeated factor. Therefore, the partial fraction decomposition will include terms for each power from 1 up to 3 of the factor \((x+1)\). The decomposition will look like \( \frac{x^2+4}{(x+1)^3} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \).

Step 2: Set Up the Equation

Start by equating the given fraction to the partial fraction decomposition: \(\frac{x^2 + 4}{(x+1)^3} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3}.\)

Step 3: Clear the Denominators

Multiply both sides of the equation by \((x+1)^3\) to eliminate the denominators. This will give: \[ x^2 + 4 = A(x+1)^2 + B(x+1) + C. \]

Step 4: Expand and Simplify the Equation

Expand the terms on the right side:- \(A(x+1)^2 = A(x^2 + 2x + 1) = Ax^2 + 2Ax + A\),- \(B(x+1) = Bx + B\).Substitute back:\[ x^2 + 4 = Ax^2 + 2Ax + A + Bx + B + C. \]

Step 5: Collect Like Terms

Group all terms with the same powers of \(x\):\[ x^2 + 4 = (A)x^2 + (2A + B)x + (A + B + C). \]

Step 6: Compare Coefficients

Match the coefficients from both sides of the equation:1. Coefficient of \(x^2\): \( A = 1 \).2. Coefficient of \(x\): \( 2A + B = 0 \).3. Constant term: \( A + B + C = 4 \).

Step 7: Solve the System of Equations

Using the coefficients:1. From \(A = 1\), substitute into the other equations.2. \(2(1) + B = 0\) becomes \(B = -2\).3. Substitute \(A = 1\) and \(B = -2\) into \(1 - 2 + C = 4\), which simplifies to \(C = 5\).

Step 8: Write the Partial Fraction Decomposition

Substitute \(A\), \(B\), and \(C\) back into the partial fraction decomposition:\[ \frac{x^2 + 4}{(x+1)^3} = \frac{1}{x+1} - \frac{2}{(x+1)^2} + \frac{5}{(x+1)^3}. \]

Key Concepts

Repeated Factors in Denominators

In partial fraction decomposition, handling repeated factors in denominators requires special care compared to distinct linear factors. A repeated factor means that the same polynomial factor appears more than once in the denominator of a rational expression, raised to a power greater than one.
For example, in the expression \( \frac{x^2+4}{(x+1)^3} \), the denominator \((x+1)^3\) is a repeated factor, repeated three times.

To perform a partial fraction decomposition with repeated factors, each successive power of the factor must be considered:

• The first term relates to the factor raised to the first power.
• The next term involves the factor raised to the second power.
• This pattern continues up to the full power present in the denominator.

Thus for the given example, the decomposition setup is \( \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \). This approach ensures that all possible contributions from the repeated factor are considered.

Polynomial Long Division

When handling rational expressions, sometimes the degree of the numerator is equal to or greater than the degree of the denominator. In such cases, polynomial long division is a vital tool for simplifying the expression.
Polynomial long division works similarly to numerical long division. You divide the highest degree terms, subtract, and continue with the next terms.

Although the given exercise \( \frac{x^2 + 4}{(x+1)^3} \) does not require division since the numerator degree is less than the denominator, understanding this concept is crucial.

• Identify the highest power in both numerator and denominator.
• Divide terms with the highest power first.
• Subtract the result from the original expression.
• Repeat the process with the next highest power of terms.

Even if not needed here, this skill is essential if degrees of numerator and denominator are equal or numerator is larger.

Solving Systems of Equations

After setting up the partial fraction expansion, the next step often involves solving a system of equations to find the unknown coefficients. This is a crucial step to ensure that the decomposed form matches the original rational expression.

For instance, once the partial fractions \( \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \) are set up, we equate and expand both sides to form an equation like \( x^2 + 4 = Ax^2 + 2Ax + A + Bx + B + C \). This scenario leads to matching coefficients:

• Match coefficients of \(x^2\): \(A = 1\)
• Match coefficients of \(x\): \(2A + B = 0\)
• Match constant terms: \(A + B + C = 4\)

This creates a small system of linear equations. Solving it involves simple algebra:
Substitute known values to find others, here resulting in \(B = -2\) and \(C = 5\). Understanding how to solve these linear systems is essential in completing the partial fraction decomposition process.