Problem 28 - College Algebra

Question

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$\frac{4 x^{2}+55 x+25}{5 x(3 x+5)^{2}}$$

Step-by-Step Solution

Answer

\( \frac{1}{5x} - \frac{1}{3(3x+5)} + \frac{32}{3(3x+5)^2} \)

Step 1: Identify the Denominator

Identify the repeating linear factors in the denominator. Here, the denominator is \(5x(3x+5)^2\). It consists of a linear factor \(5x\) and a repeated linear factor \((3x+5)^2\).

Step 2: Write the Partial Fraction Decomposition Form

For the repeated linear factor \((3x+5)\), we write terms for each power up to the power it is raised: \(\frac{A}{5x} + \frac{B}{3x+5} + \frac{C}{(3x+5)^2}\). This is the general form for partial fraction decomposition with repeating linear factors.

Step 3: Create an Equation from the Numerators

Set up an equation by equating the numerator of the original fraction to the combined numerators from the partial fractions: \[ 4x^2 + 55x + 25 = A(3x+5)^2 + B(5x)(3x+5) + C(5x). \]

Step 4: Expand and Combine Terms

Expand the terms: \( A(3x+5)^2 = A(9x^2 + 30x + 25) \), \( B(5x)(3x+5) = B(15x^2 + 25x) \), and \( C(5x) = 5Cx \). Combine all on the right-hand side: \[ 9Ax^2 + 30Ax + 25A + 15Bx^2 + 25Bx + 5Cx. \]

Step 5: Match Coefficients

Match coefficients of corresponding powers of \(x\) from both sides of the equation: - For \(x^2\): \(9A + 15B = 4\)- For \(x\): \(30A + 25B + 5C = 55\)- For the constant term: \(25A = 25\).Solve these equations to find \(A\), \(B\), and \(C\).

Step 6: Solve for Coefficients

From \(25A = 25\), we find \(A = 1\). Substitute \(A\) in the other two:- \(9 \times 1 + 15B = 4\), which gives \(15B = -5\) so \(B = -\frac{1}{3}\).- \(30 \times 1 + 25(-\frac{1}{3}) + 5C = 55\), simplify and solve to get \(C = \frac{32}{3}\).

Step 7: Write the Final Decomposition

Insert the values of \(A\), \(B\), and \(C\) back into the partial fraction form: \[ \frac{1}{5x} - \frac{1}{3(3x+5)} + \frac{32}{3(3x+5)^2}. \] This is the partial fraction decomposition.

Key Concepts

Linear Factors

In partial fraction decomposition, linear factors are essential components. A linear factor is a polynomial of degree one, typically written in the form \(ax + b\). It looks like a straight line when graphed, hence the name 'linear'. Recognizing linear factors in an expression is the first step towards decomposition. For instance, in the problem at hand, \(5x\) and \((3x + 5)\) are linear factors.

These factors can significantly influence the way we set up the partial fractions. Here’s why linear factors are crucial:

• Linear factors determine the number of terms needed in the partial fraction decomposition.
• Each distinct linear factor contributes a separate term, which simplifies solving equations later.

Understanding linear factors allows us to break down complex expressions into more manageable parts. It’s key for further steps in partial fraction decomposition.

Repeating Factors

Repeating factors in an algebraic expression occur when a linear factor is raised to a power greater than one. For the given problem, \((3x + 5)^2\) is a repeated linear factor. Special treatment is necessary when dealing with such factors as they affect the partial fraction setup.

Here's how repeating factors impact the decomposition:

• Each level of the repeated factor requires its own term in the partial fraction expansion. For example, \((3x + 5)^2\) leads us to include separate terms for \(\frac{B}{3x+5}\) and \(\frac{C}{(3x+5)^2}\).
• This ensures each power of the factor is accounted for, preventing any loss of information.

Handling repeating factors correctly is crucial for accurate decomposition, ensuring the entire expression is represented.

Partial Fractions

Partial fractions are a way of expressing a complicated fraction into simpler, more digestible pieces. The process involves breaking down a single complex fraction into a sum of simpler fractions, which are often easier to integrate or differentiate in calculus.

For the exercise, the fraction \(\frac{4x^2+55x+25}{5x(3x+5)^2}\) is decomposed into partial fractions. This results in the expression:

• \(\frac{1}{5x} - \frac{1}{3(3x+5)} + \frac{32}{3(3x+5)^2}\)

Decomposition into partial fractions involves several steps:

• Identify and write down the forms based on linear and repeating factors.
• Solve for unknown coefficients by matching terms and simplifying.

Mastering partial fractions simplifies solving algebra problems and is a useful tool in many mathematical fields.

Algebra Problem Solving

Solving algebra problems often requires breaking down complex expressions into simpler components. Partial fraction decomposition is one such technique that serves this purpose well. It’s highly useful in solving integrals, differential equations, and various algebraic equations.

Key steps in algebra problem solving with partial fractions include:

• Identifying and isolating factors—both linear and repeating.
• Writing the partial fraction form based on these factors.
• Creating and solving equations for coefficients by matching like terms.

Approaching algebra problems with a clear, structured method like partial fraction decomposition makes complex solutions accessible and understandable. It breaks down the intimidating process into smaller, logical steps, boosting the student's confidence in solving such problems. Every algebra student should feel empowered by mastering these methods to tackle and solve challenging equations efficiently.