Find The Partial Fraction Decomposition For The Following Rational Expression

Finding the Partial Fraction Decomposition of Rational Expressions: A Comprehensive Guide

Partial fraction decomposition is a crucial technique in calculus and other areas of mathematics, particularly when dealing with integrals of rational functions. This process involves breaking down a complex rational expression (a fraction where the numerator and denominator are polynomials) into simpler fractions, making integration and other manipulations significantly easier. This comprehensive guide will walk you through the process, covering various cases and offering practical examples.

Understanding Rational Expressions and Partial Fractions

A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, (3x² + 2x + 1) / (x³ - x² - 2x) is a rational expression. We aim to decompose this into a sum of simpler fractions called partial fractions. The general form of a partial fraction decomposition depends on the factors of the denominator.

Types of Factors in the Denominator and Their Corresponding Partial Fractions

The key to successful partial fraction decomposition lies in understanding the factors of the denominator polynomial. The form of the partial fractions depends directly on the nature of these factors:

1. Distinct Linear Factors

If the denominator has distinct linear factors (e.g., (x-a)(x-b)(x-c)), then the partial fraction decomposition takes the form:

A/(x-a) + B/(x-b) + C/(x-c) + ...

where A, B, C, etc., are constants that need to be determined.

Example:

Let's decompose 3x + 11 / (x-1)(x+4). The distinct linear factors are (x-1) and (x+4). Therefore, the partial fraction decomposition will be:

A/(x-1) + B/(x+4)

To find A and B, we combine the fractions on the right-hand side, then equate the coefficients of the corresponding powers of x in the numerators:

A/(x-1) + B/(x+4) = (A(x+4) + B(x-1)) / ((x-1)(x+4))

Equating this to (3x + 11) / (x-1)(x+4) gives:

A(x+4) + B(x-1) = 3x + 11

Expanding and collecting terms:

(A+B)x + (4A - B) = 3x + 11

Comparing coefficients, we get the system of equations:

A + B = 3
4A - B = 11

Solving this system (e.g., by substitution or elimination), we find A = 2 and B = 1. Therefore, the partial fraction decomposition is:

2/(x-1) + 1/(x+4)

2. Repeated Linear Factors

If the denominator has repeated linear factors (e.g., (x-a)²(x-b)), the partial fraction decomposition includes terms for each power of the repeated factor:

A/(x-a) + B/(x-a)² + C/(x-b) + ...

Example:

Consider (x² + 1) / (x-1)²(x+2). The denominator has a repeated linear factor (x-1)² and a distinct linear factor (x+2). The partial fraction decomposition is:

A/(x-1) + B/(x-1)² + C/(x+2)

Similar to the previous example, combining the fractions, equating coefficients, and solving the resulting system of equations will yield the values of A, B, and C.

3. Irreducible Quadratic Factors

If the denominator contains irreducible quadratic factors (quadratic factors that cannot be factored into real linear factors, e.g., x² + 1), the corresponding partial fraction will have a linear numerator:

(Ax + B)/(x² + 1)

Example:

Decomposing (3x² + 2x + 1) / (x(x² + 1)), we have a linear factor (x) and an irreducible quadratic factor (x² + 1):

A/x + (Bx + C) / (x² + 1)

Again, combining fractions, equating coefficients, and solving the system of equations will give you the values of A, B, and C.

4. Repeated Irreducible Quadratic Factors

If the denominator contains repeated irreducible quadratic factors (e.g., (x² + 1)²), each power of the repeated factor will have a linear numerator:

(Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + ...

Steps for Partial Fraction Decomposition

Here's a step-by-step guide summarizing the process:

1. Factor the denominator: Completely factor the denominator polynomial into linear and irreducible quadratic factors.

2. Write the general form: Based on the types of factors identified in step 1, write the general form of the partial fraction decomposition. This involves assigning constants (A, B, C, etc.) to each term.

3. Combine the fractions: Combine the partial fractions on the right-hand side into a single fraction.

4. Equate coefficients: Equate the numerator of the combined fraction to the original numerator of the rational expression. This will give you a system of equations involving the constants A, B, C, etc.

5. Solve the system of equations: Solve the system of equations to find the values of the constants. Substitution, elimination, or matrix methods can be used.

6. Substitute the values: Substitute the values of the constants back into the general form of the partial fraction decomposition.

Advanced Techniques and Considerations

• Heaviside Cover-up Method: For distinct linear factors, the Heaviside cover-up method can simplify the process of finding the constants. To find the constant associated with a linear factor (x-a), cover up the (x-a) term in the denominator and substitute x = a into the remaining expression.

• Complex Numbers: If the denominator contains irreducible quadratic factors with complex roots, the partial fractions will involve complex numbers.

• Long Division: If the degree of the numerator is greater than or equal to the degree of the denominator, you must perform polynomial long division before proceeding with partial fraction decomposition. This will result in a polynomial plus a proper rational function (where the degree of the numerator is less than the degree of the denominator).

Conclusion

Partial fraction decomposition is a powerful technique with wide-ranging applications in mathematics. Mastering this technique requires a solid understanding of polynomial factoring and the ability to solve systems of equations. By following the steps outlined above and practicing with various examples, you can confidently decompose complex rational expressions into simpler forms suitable for integration, solving differential equations, and other advanced mathematical manipulations. Remember to always thoroughly check your work by recombining the partial fractions to ensure you arrive back at the original rational expression. This thoroughness will prevent errors and ensure accuracy in your calculations.