Partial Fraction Decomposition With Long Division

Partial Fraction Decomposition: Mastering the Art with Long Division

Partial fraction decomposition is a crucial technique in calculus and other branches of mathematics, used to simplify complex rational functions into simpler, more manageable expressions. This process often involves a preliminary step: long division. While not always necessary, long division ensures that the degree of the numerator is less than the degree of the denominator – a prerequisite for applying the standard partial fraction decomposition techniques. This comprehensive guide will walk you through the entire process, covering both long division and partial fraction decomposition in detail, with numerous examples to solidify your understanding.

When Do You Need Long Division?

Before diving into the mechanics, let's clarify when long division is required. Consider a rational function of the form:

P(x) / Q(x)

where P(x) is the numerator and Q(x) is the denominator. Long division is necessary if the degree of P(x) is greater than or equal to the degree of Q(x). If the degree of P(x) is less than the degree of Q(x), you can proceed directly to partial fraction decomposition.

Example:

Let's consider the rational function:

(x³ + 2x² + 4x + 8) / (x² + 4)

Here, the degree of the numerator (3) is greater than the degree of the denominator (2). Therefore, we must perform long division before proceeding with partial fraction decomposition.

Performing Long Division

Long division for polynomials follows a similar algorithm to long division for numbers. Let's illustrate with the example above:

                  x + 2
        _________________________
x² + 4 | x³ + 2x² + 4x + 8
         - (x³       + 4x)
         _________________________
                 2x²          + 8
                 -(2x²       + 8)
                 _________________________
                           0

This shows us that:

(x³ + 2x² + 4x + 8) / (x² + 4) = x + 2

Therefore, we've successfully simplified the original rational function using long division. In cases where there's a remainder, the remainder will be a new numerator over the original denominator.

Example with a Remainder:

Let's consider:

(2x³ + x² + 3x + 1) / (x² - 1)

                2x + 1
        _________________________
x² - 1 | 2x³ + x² + 3x + 1
         - (2x³     - 2x)
         _________________________
                x² + 5x + 1
                - (x²     - 1)
                _________________________
                       5x + 2

This gives us:

(2x³ + x² + 3x + 1) / (x² - 1) = 2x + 1 + (5x + 2) / (x² - 1)

Now we can apply partial fraction decomposition to the remaining fraction (5x + 2) / (x² - 1).

Partial Fraction Decomposition: The Core Technique

After performing long division (if necessary), we are left with a proper rational function (degree of the numerator < degree of the denominator). Now, we can apply partial fraction decomposition. This involves breaking down the rational function into a sum of simpler fractions. The exact form depends on the factors of the denominator:

Case 1: Distinct Linear Factors

If the denominator can be factored into distinct linear factors, the partial fraction decomposition will take the form:

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

where A, B, C, etc., are constants to be determined and a, b, c are the distinct roots of Q(x).

Example:

(3x + 5) / (x - 1)(x + 2) = A/(x - 1) + B/(x + 2)

We can solve for A and B by multiplying both sides by (x - 1)(x + 2) and then substituting values of x that eliminate variables.

Case 2: Repeated Linear Factors

If the denominator has repeated linear factors, we need to include terms for each power of the repeated factor:

(P(x) / Q(x)) = A/(x - a) + B/(x - a)² + C/(x - a)³ + ...

Example:

(2x + 1) / (x - 3)² = A/(x - 3) + B/(x - 3)²

Case 3: Irreducible Quadratic Factors

If the denominator contains irreducible quadratic factors (quadratic factors that cannot be factored further into real linear factors), we use a linear numerator for each quadratic factor:

(P(x) / Q(x)) = (Ax + B)/(x² + px + q) + ...

Example:

(x² + 1) / (x² + 1)(x + 2) = (Ax + B) / (x² + 1) + C/(x + 2)

This factor is particularly relevant when dealing with complex roots.

Case 4: Repeated Irreducible Quadratic Factors

If there are repeated irreducible quadratic factors, we again include terms for each power of the repeated factor, each with a linear numerator:

(P(x) / Q(x)) = (Ax + B)/(x² + px + q) + (Cx + D)/(x² + px + q)² + ...

Solving for the Constants

Once you have set up the partial fraction decomposition, you need to solve for the unknown constants (A, B, C, etc.). There are two main methods:

1. Equating Coefficients: After clearing the denominators, expand the equation and equate the coefficients of corresponding powers of x. This results in a system of linear equations that can be solved simultaneously.

2. Substituting Values: Substitute specific values of x (often the roots of the denominator) to simplify the equation and solve for the constants more directly.

Complete Example Combining Long Division and Partial Fraction Decomposition

Let's solve:

(x⁴ + 2x³ + 3x² + 4x + 5) / (x² + x + 1)

1. Long Division:

                 x³ + x² + 2x + 2
       _________________________
x² + x + 1 | x⁴ + 2x³ + 3x² + 4x + 5
            - (x⁴ + x³ + x²)
            _________________________
                    x³ + 2x² + 4x + 5
                    -(x³ + x² + x)
                    _________________________
                         x² + 3x + 5
                         -(x² + x + 1)
                         _________________________
                                 2x + 4

This simplifies to:

x³ + x² + 2x + 2 + (2x + 4) / (x² + x + 1)

2. Partial Fraction Decomposition:

The quadratic x² + x + 1 is irreducible. Thus we use a linear numerator.

(2x + 4) / (x² + x + 1) = (Ax + B) / (x² + x + 1)

Multiplying both sides by (x² + x + 1) gives:

2x + 4 = Ax + B

Equating coefficients:

• A = 2
• B = 4

3. Final Result:

The final decomposition is:

x³ + x² + 2x + 2 + (2x + 4) / (x² + x + 1)

Conclusion

Partial fraction decomposition, often used in conjunction with long division, is a fundamental technique for simplifying complex rational functions. By mastering both long division and the various cases of partial fraction decomposition, you’ll gain a powerful tool for solving problems in calculus and beyond. Remember to carefully consider the factors of the denominator and select the appropriate form for your partial fractions. Practice is key to mastering this technique – work through many examples to build your confidence and proficiency. With consistent practice and a strong understanding of the underlying principles, you'll confidently navigate the world of partial fraction decomposition.