Find the partial fraction decomposition for each rational expression. 5-2x / (x² + 2)(x - 1)

Verified step by step guidance

1. Identify the form of the partial fraction decomposition. Since the denominator is \( (x^2 + 2)(x - 1) \), where \( x^2 + 2 \) is an irreducible quadratic and \( x - 1 \) is a linear factor, the decomposition will be of the form: \[ \frac{5 - 2x}{(x^2 + 2)(x - 1)} = \frac{Ax + B}{x^2 + 2} + \frac{C}{x - 1} \] where \( A \), \( B \), and \( C \) are constants to be determined.
2. Multiply both sides of the equation by the common denominator \( (x^2 + 2)(x - 1) \) to clear the fractions: \[ 5 - 2x = (Ax + B)(x - 1) + C(x^2 + 2) \]. This step eliminates the denominators and allows us to work with polynomials.
3. Expand the right-hand side by distributing: \[ (Ax + B)(x - 1) = Ax^2 - Ax + Bx - B \] and \[ C(x^2 + 2) = Cx^2 + 2C \]. Combine these to get: \[ 5 - 2x = (A + C)x^2 + (-A + B)x + (-B + 2C) \].
4. Equate the coefficients of corresponding powers of \( x \) from both sides. On the left, the polynomial is \( 5 - 2x = 0x^2 - 2x + 5 \). So, set up the system: \[ \begin{cases} A + C = 0 \\ -A + B = -2 \\ -B + 2C = 5 \end{cases} \].
5. Solve the system of equations for \( A \), \( B \), and \( C \). Once these constants are found, substitute them back into the partial fraction form \( \frac{Ax + B}{x^2 + 2} + \frac{C}{x - 1} \) to complete the decomposition.

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 the denominator into factors and assigning unknown constants to each fraction.

Factoring the Denominator

Factoring the denominator is essential to identify the distinct linear and quadratic factors that determine the form of the partial fractions. In this problem, the denominator is already factored as (x^2 + 2)(x - 1), where x^2 + 2 is an irreducible quadratic and x - 1 is a linear factor.

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of partial fractions with unknown coefficients, you multiply both sides by the denominator to clear fractions. Then, equate coefficients of corresponding powers of x or substitute convenient values of x to form a system of equations, which you solve to find the unknown constants.