Partial Fraction Decomposition of x / (x1(x-1)x2)

Partial fraction decomposition is a technique in algebra and calculus used to break down a complex rational expression into simpler fractions. This method is particularly useful for integrating or simplifying expressions. In this article, we will explore the process of decomposing the expression

Decomposition Process

The given expression is:

$$ frac{x}{x1(x-1)x2} frac{A}{x1} frac{B}{x-1} frac{C}{x2} $$

This decomposition can be verified as follows:

Multiply both sides by the denominator (x1(x-1)x2):

$$ x A(x-1)x2 Bx1x2 Cx1x-1 $$

Solving for Constants

To find the constants (A), (B), and (C), we can substitute specific values for (x). Let's go through the process step-by-step.

Substituting x -1

If (x -1), the equation becomes:

$$ -1 A((-1)-1)(-1)2 B(x1)(-1)2 C(x1)((-1)-1) $$

Simplify the equation:

$$ -1 -2A $$

Solving for (A):

$$ A frac{1}{2} $$

Substituting x 1

If (x 1), the equation becomes:

$$ 1 A(1-1)(1)2 B(x1)(1)2 C(x1)(1-1) $$

Simplify the equation:

$$ 1 6B $$

Solving for (B):

$$ B frac{1}{6} $$

Substituting x -2

If (x -2), the equation becomes:

$$ -2 A(-2-1)(-2)2 B(x1)(-2)2 C(x1)(-2-1) $$

Simplify the equation:

$$ -2 3C $$

Solving for (C):

$$ C -frac{2}{3} $$

Final Decomposition

With the values of (A), (B), and (C) determined, we can write the full decomposition as:

$$ frac{x}{x1(x-1)x2} frac{1}{2} cdot frac{1}{x1} frac{1}{6} cdot frac{1}{x-1} - frac{2}{3} cdot frac{1}{x2} $$

This further simplifies to:

$$ frac{x}{x1(x-1)x2} frac{1}{2x1} frac{1}{6(x-1)} - frac{2}{3x2} $$

Conclusion

This decomposition allows us to break down the given rational expression into simpler fractions, making it easier to work with in various mathematical operations and integrations. Understanding partial fraction decomposition is crucial for advanced algebra and calculus.