Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{5 x+14}{2 x^{2}+12 x+18} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Denominator

First, we need to factorize the quadratic expression in the denominator: \(2x^2 + 12x + 18\). Begin by finding the greatest common factor, which is 2: \(2(x^2 + 6x + 9)\). Notice that \(x^2 + 6x + 9\) can be further factored as a perfect square: \((x+3)^2\). Therefore, the denominator becomes \(2(x+3)^2\).

Step 2: Set Up Partial Fraction Decomposition

Now that the denominator is \(2(x+3)^2\), the partial fraction decomposition can be set up as \(\frac{A}{x+3} + \frac{B}{(x+3)^2}\). This assumes A and B are constants to be determined.

Step 3: Clear the Denominator

Multiply both sides by the common denominator \(2(x+3)^2\) to eliminate the fractions: \(5x + 14 = A(x+3) + B\).

Step 4: Expand and Collect Terms

Expand the right side: \(Ax + 3A + B\). Collect terms to give the equation \(Ax + (3A + B)\).

Step 5: Solve for A and B

Set up the equations by comparing coefficients:1. From \(Ax = 5x\), we get \(A = 5\).2. From \(3A + B = 14\), substitute \(A = 5\) to get \(3(5) + B = 14\), which simplifies to \(15 + B = 14\), so \(B = -1\).

Step 6: Write the Decomposition

Substituting the values found for A and B, the partial fraction decomposition of \(\frac{5x + 14}{2(x+3)^2}\) is:\[\frac{5}{2(x+3)} - \frac{1}{2(x+3)^2}.\]

Key Concepts

Repeating Linear Factors

Repeating linear factors appear frequently in partial fraction decomposition and they're essential to mastering this mathematical technique. In this context, a repeating linear factor is an expression that has the same base term repeated; for example,

• (x+3) occurs multiple times in the denominator of the given rational expression,

Effectively handling repeating linear factors involves setting up a distinct decomposition for each repetition. Hence, when facing

• (x+3)^2, you need to apply a separate breakdown:
  • one for (x+3) as \( \frac{A}{x+3} \)
  • and another for (x+3)^2 as \( \frac{B}{(x+3)^2} \).

This layered setup allows us to solve for each section's constant separately, making the problem more navigable. Identifying and addressing repeating linear factors correctly is crucial for accurate results.

Greatest Common Factor

The greatest common factor (GCF) in polynomial factorization greatly simplifies expressions. Consider it the biggest factor you can pull out from each term of a polynomial. In our exercise, the expression in the denominator,

• 2x^2 + 12x + 18, is a polynomial that shares a common factor: 2.

By extracting this GCF, we transform the polynomial into a simpler form:

• 2(x^2 + 6x + 9).

This step is crucial because it immediately simplifies the entire decomposition process. The reduced expression

• (x^2 + 6x + 9)

can often be further factorized, paving the way for determining the partial fraction decomposition more efficiently. Remember, identifying and using the GCF can drastically reduce the complexity of your calculations.

Perfect Square

Understanding perfect squares can ease difficulties with factorization. A perfect square trinomial is a polynomial that can be expressed as the square of a binomial. In other words, it's an expression of the form

• a^2 + 2ab + b^2 = (a+b)^2.

In our exercise,

• x^2 + 6x + 9

represents a perfect square because it can be rewritten as:

• (x + 3)^2.

Recognizing perfect squares allows us to considerably simplify expressions, like reducing the polynomial in the denominator. This recognition made it possible to reduce

• 2(x^2 + 6x + 9) to 2(x+3)^2

which significantly simplifies the decomposition steps. Identify perfect squares to avoid excessive calculation and streamline the problem-solving process.

Quadratic Expression Factorization

Factorizing quadratic expressions is a crucial tool for simplifying expressions in partial fraction decomposition. The goal is to transform a second-degree polynomial into a product of binomials.

• The expression x^2 + 6x + 9 in our exercise represents a quadratic expression.

Through factorization, it becomes:

• (x + 3)(x + 3) or (x + 3)^2.

Factorization involves finding two values that multiply to the constant term and add to the linear coefficient. Recognizing that x^2 + 6x + 9 is a perfect square helps illustrate this process. Breaking down the polynomial helps build a clear pathway towards partial fraction decomposition. Accurately factorizing quadratic expressions simplifies solving complex algebraic equations and provides a cleaner path to the final solution.