Write the partial fraction decomposition of each rational expression. x²/(x − 1)² (x + 1)

Verified step by step guidance

1. Identify the denominator and its factors. The denominator is \((x - 1)^2 (x + 1)\), which consists of a repeated linear factor \((x - 1)^2\) and a distinct linear factor \((x + 1)\).
2. Set up the form of the partial fraction decomposition. For the repeated linear factor \((x - 1)^2\), include terms with denominators \((x - 1)\) and \((x - 1)^2\). For the linear factor \((x + 1)\), include a term with denominator \((x + 1)\). So, write the decomposition as: \[\frac{x^2}{(x - 1)^2 (x + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 1}\]
3. Multiply both sides of the equation by the common denominator \((x - 1)^2 (x + 1)\) to clear the denominators. This gives: \[x^2 = A(x - 1)(x + 1) + B(x + 1) + C(x - 1)^2\]
4. Expand the right-hand side by multiplying out each term: - Expand \(A(x - 1)(x + 1)\) using the difference of squares formula. - Expand \(B(x + 1)\). - Expand \(C(x - 1)^2\) by squaring the binomial.
5. Collect like terms on the right-hand side and equate the coefficients of corresponding powers of \(x\) from both sides. This will give a system of equations to solve for \(A\), \(B\), and \(C\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or solving equations. It involves breaking down a fraction into components with denominators that are factors of the original denominator.

Repeated Linear Factors

When the denominator contains repeated linear factors, such as (x - 1)², the decomposition must include terms for each power of the repeated factor. For example, for (x - 1)², the decomposition includes terms with denominators (x - 1) and (x - 1)², each with its own constant numerator.

Setting Up and Solving Equations for Coefficients

After expressing the rational function as a sum of partial fractions, you multiply both sides by the common denominator to clear fractions. Then, by equating coefficients of corresponding powers of x or substituting convenient values, you solve for the unknown constants in the numerators.