partial fraction decomposition practice

Guri Bee

2 min read

·

21-10-2024

1

0

Share

Mastering Partial Fraction Decomposition: Practice Problems and Solutions

Partial fraction decomposition is a powerful technique used in calculus and other areas of mathematics to break down complex rational expressions into simpler fractions. This process is particularly useful for integrating rational functions and solving differential equations.

Understanding the Basics

Before diving into practice problems, let's recap the core idea behind partial fraction decomposition. We aim to express a rational function, where the degree of the numerator is less than the degree of the denominator, as a sum of simpler fractions. These simpler fractions have linear or quadratic denominators.

The Key Steps

1. Factor the denominator: First, factor the denominator of the original rational expression into linear and quadratic factors.

2. Set up the partial fractions: For each linear factor (ax+b), introduce a term of the form A/(ax+b). For each quadratic factor (ax^2+bx+c), introduce a term of the form (Bx+C)/(ax^2+bx+c).

3. Solve for the unknown coefficients: Multiply both sides of the equation by the original denominator and then substitute values for x that make individual terms zero. This helps isolate the unknown coefficients.

Practice Problems with Solutions

Let's solidify our understanding with some practice problems, drawing inspiration from discussions on GitHub.

Problem 1: Decompose the rational expression (2x+1)/(x^2-1) into partial fractions.

Solution:

Step 1: Factor the denominator: (x^2-1) = (x+1)(x-1)

Step 2: Set up the partial fractions: (2x+1)/(x^2-1) = A/(x+1) + B/(x-1)

Step 3: Multiply both sides by (x+1)(x-1): 2x+1 = A(x-1) + B(x+1)

Step 4: Solve for A and B. Let's try substituting x = 1 and x = -1:

• For x = 1: 3 = 2B => B = 3/2
• For x = -1: -1 = -2A => A = 1/2

Therefore, the partial fraction decomposition is:

(2x+1)/(x^2-1) = (1/2)/(x+1) + (3/2)/(x-1)

Problem 2: Decompose the rational expression (x^2+1)/(x3-x) into partial fractions.

Solution:

Step 1: Factor the denominator: x^3 - x = x(x^2 - 1) = x(x+1)(x-1)

Step 2: Set up the partial fractions: (x^2+1)/(x3-x) = A/x + B/(x+1) + C/(x-1)

Step 3: Multiply both sides by x(x+1)(x-1): x^2 + 1 = A(x+1)(x-1) + Bx(x-1) + Cx(x+1)

Step 4: Solve for A, B, and C. Let's substitute x = 0, x = -1, and x = 1:

• For x = 0: 1 = -A => A = -1
• For x = -1: 2 = 2B => B = 1
• For x = 1: 2 = 2C => C = 1

Therefore, the partial fraction decomposition is:

(x^2+1)/(x3-x) = -1/x + 1/(x+1) + 1/(x-1)

Additional Insights

• Repeated Factors: If a factor appears multiple times in the denominator, we introduce a term for each power of that factor. For example, (1/x^3) would be decomposed as A/x + B/x^2 + C/x^3.

• Irreducible Quadratic Factors: When dealing with quadratic factors that cannot be factored further (e.g., x^2 + 1), we use the form (Bx+C)/(ax^2+bx+c).

• Integration: The main advantage of partial fraction decomposition is that it simplifies integration. The individual terms resulting from the decomposition are often easier to integrate using standard techniques.

Conclusion

Practice is key to mastering partial fraction decomposition. By working through various problems and analyzing different scenarios, you can gain confidence in applying this powerful technique to solve complex mathematical problems. The GitHub community provides a valuable resource for exploring problems and solutions, facilitating your learning journey.