Partial Fraction Decomposition for Rational Functions

Partial fraction decomposition is a valuable tool in algebra and calculus, simplifying complex rational functions into a sum of simpler rational expressions. This article delves into the process of decomposing a specific rational function, offering a clear, step-by-step guide that follows the principles of algebra.

Understanding the Rational Function

The rational function in question is given as:

frac{x^2 - 29x - 5}{x - 4^2x^2 3} end{latex}

The goal is to decompose this into a sum of simpler fractions.

Step-by-Step Solution

Let's decompose the given rational function using the method of partial fractions:

frac{x^2 - 29x - 5}{x - 4^2x^2 3} frac{A}{x - 4} frac{B}{x - 4^2} frac{Cx D}{x^2 3} end{latex}

Step 1: Setting up the Equation

Multiply both sides by the common denominator to clear the fractions:

(x - 4^2x^2 3)(frac{A}{x - 4} frac{B}{x - 4^2} frac{Cx D}{x^2 3}) x^2 - 29x - 5 end{latex>

Step 2: Simplifying

After simplifying, we get the equation:

x^2 - 29x - 5 Ax - 4x^2 3 Bx^2 3 (Cx D)x^2 3 end{latex>

Step 3: Collecting Like Terms

Collect the coefficients of each power of x:

Constant Term: 5 -12A 3B 16D Co-efficient of x3: 0 -4A B - 8C D Co-efficient of x2: 1 -4A B - 8C D Co-efficient of x: -29 -3A 16C - 8D

Step 4: Solving the System of Equations

From the simplified equations:

-12A 3B 16D 5 -4A B - 8C D 1 -29 -13A - 8D 1 -3A - 16C 4D

We solve for A, B, C, D. We start with B:

Substitute B -5 from the equation for the constant term.

Use A -C from the coefficient of x3.

Use the simplified coefficient of x2:

-4A - 5 - 8C D 1

4A D 6

Use the coefficient of x:

-29 -13A - 8D

5 -3A - 4D

Solving these, we find:

A 1

D 2

C -A -1

Finally, we get:

Cx D -x 2

Conclusion

The decomposed form of the given rational function is:

frac{x^2 - 29x - 5}{x - 4^2x^2 3} frac{1}{x - 4} - frac{5}{x - 4^2} frac{2 - x}{x^2 3} end{latex}