Express the rational function as a sum or difference or simpler rational expressions.
$\frac{4}{x^3-3x^2-x+3}$

• A. $\frac{1}{2(x-3)}+\frac{1}{2(x+1)}-\frac{1}{x-1}$
• B. $\frac{4}{x^3}-\frac{4}{3x^2}-\frac{4}{x+3}$
• C. $\frac{4}{x^3}-\frac{4}{3x^2}+\frac{4}{x+3}$
• D. $\frac{1}{2(x-3)}-\frac{1}{2(x+1)}+\frac{1}{x-1}$

Answer: (A)

Verified step by step guidance

1. Factorize the denominator of the given rational function \( \frac{4}{x^3 - 3x^2 - x + 3} \). Use grouping to rewrite the denominator as \( (x^2(x - 3)) - (1(x - 3)) \), which simplifies to \( (x^2 - 1)(x - 3) \). Further factorize \( x^2 - 1 \) as \( (x - 1)(x + 1) \). The fully factorized denominator is \( (x - 3)(x - 1)(x + 1) \).
2. Rewrite the rational function as \( \frac{4}{(x - 3)(x - 1)(x + 1)} \). To express this as a sum of simpler fractions, use partial fraction decomposition: \( \frac{4}{(x - 3)(x - 1)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x - 1} + \frac{C}{x + 1} \), where \( A \), \( B \), and \( C \) are constants to be determined.
3. Multiply through by the denominator \( (x - 3)(x - 1)(x + 1) \) to eliminate the fractions: \( 4 = A(x - 1)(x + 1) + B(x - 3)(x + 1) + C(x - 3)(x - 1) \). Expand each term on the right-hand side.
4. Group like terms and equate coefficients of \( x^2 \), \( x \), and the constant term on both sides of the equation. Solve the resulting system of equations to find the values of \( A \), \( B \), and \( C \).
5. Substitute the values of \( A \), \( B \), and \( C \) back into the partial fraction decomposition to express the original rational function as a sum of simpler rational expressions. Simplify each term if necessary.