Problem 27 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{x^{2}}{(x-1)^{2}(x+1)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{x^{2}}{(x-1)^{2}(x+1)}\) is \(-\frac{0.5}{x-1} + \frac{0.5}{(x-1)^2} + \frac{0.5}{x+1}\)

Step 1: Express the Fraction in Partial Fraction Form

Given that the denominator has two distinct factors of which \(x-1\) is a repeated root, express the rational function as the sum of partial fractions as follows:\n\[\frac{x^{2}}{(x-1)^{2}(x+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1}\]

Step 2: Clear Fractions

Multiply through by the common denominator \((x-1)^{2}(x+1)\) to get rid of the fraction:\n\[x^{2} = A(x-1)(x+1) + B(x+1) + C(x-1)^{2}\]

Step 3: Solve for Constants

Substitute suitable values for \(x\) to solve for the constants \(A, B,\) and \(C\). For instance, when \(x = 1\), the equation simplifies to:\n\[1 = B(2)\]\nHence \(B = 0.5\). Similarly, for \(x = -1\), we have: \n\[1 = -2A\]\nMeans \(A = -0.5\). Finally, substituting \(x = 0\),\n\[0 = 0.5 - C\]\nHence \(C = 0.5\].

Step 4: Write the Final Decomposition

Substitute the solved constants \(A, B\) and \(C\) back to the partial fractions decomposition formula derived in Step 1. This gives the final answer as:\n\[\frac{x^{2}}{(x-1)^{2}(x+1)} = -\frac{0.5}{x-1} + \frac{0.5}{(x-1)^2} + \frac{0.5}{x+1}\]

Key Concepts

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. Understanding these expressions is essential for solving complex algebraic equations. They are common in various math problems, especially those involving algebraic fractions. A rational expression is written as \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). Some key points about rational expressions include:

• They represent the division of one polynomial by another.
• Simplifying involves factoring both the numerator and the denominator.
• They can be rewritten in simpler forms using partial fraction decomposition.

In the given exercise, we are working with the rational expression \( \frac{x^2}{(x-1)^2(x+1)} \), which has a polynomial numerator of degree 2 and a polynomial denominator of degree 3.

Repeated Roots

Repeated roots occur in rational expressions when a factor in the denominator is raised to a power. It means that the polynomial has the same linear factor appearing more than once. In the problem we're working with, the factor \((x-1)\) is repeated, appearing as \((x-1)^2\). Understanding repeated roots is crucial when preparing to decompose rational expressions into partial fractions.

• In partial fraction decomposition, each instance of the repeated root is handled separately.
• For a factor \((x-a)^n\), you need to include separate terms for every power: \( \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n} \).
• This step ensures that all possible contributions of the repeated factor are accounted for in the decomposition.

Dealing with repeated roots allows us to thoroughly capture the behavior of the original rational expression.

Constants Solution

To solve partial fraction decomposition, we need to determine the constants that make the decomposed expression equal to the original rational expression. These constants are critical because they transform abstract expressions into concrete numbers.

• Use substitution to solve for constants after clearing the fractions in the rational expression.
• Choose strategic values for \( x \) that simplify the expression, like using roots of the denominator.
• By substituting these values back into the equation, we obtain numbers for our constants \( A, B, \) and \( C \).

For instance, in the provided solution, setting \( x = 1 \), \( x = -1 \), and \( x = 0 \) helped us find \( A = -0.5 \), \( B = 0.5 \), and \( C = 0.5 \). These constants are then substituted back into the equation, providing the complete partial fraction decomposition.