Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x² - 3x - 4)/(x³ + x² - 2x)

Verified step by step guidance

1. First, factor the denominator completely. The denominator is \(x^3 + x^2 - 2x\). Start by factoring out the greatest common factor, which is \(x\), giving \(x(x^2 + x - 2)\).
2. Next, factor the quadratic \(x^2 + x - 2\). Find two numbers that multiply to \(-2\) and add to \(1\). These are \(2\) and \(-1\), so the factorization is \((x + 2)(x - 1)\).
3. Now, rewrite the original rational expression with the factored denominator: \(\frac{4x^2 - 3x - 4}{x(x + 2)(x - 1)}\).
4. Set up the partial fraction decomposition form. Since all factors in the denominator are linear and distinct, express it as: \(\frac{A}{x} + \frac{B}{x + 2} + \frac{C}{x - 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
5. Multiply both sides of the equation by the common denominator \(x(x + 2)(x - 1)\) to clear the denominators, resulting in an equation involving polynomials. Then, expand and collect like terms to form an equation that can be solved for \(A\), \(B\), and \(C\) by equating coefficients or by substituting convenient values of \(x\).

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 simpler denominators, typically linear or quadratic factors.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can be linear or quadratic expressions. For partial fraction decomposition, factoring the denominator completely is essential to identify the simpler fractions. For example, factoring x^3 + x^2 - 2x helps determine the denominators of the partial fractions.

Setting Up and Solving Systems of Equations

After expressing the rational expression as a sum of partial fractions, you equate the original numerator to the combined numerator of the decomposed fractions. This results in a system of equations based on coefficients of corresponding powers of x. Solving this system finds the unknown constants in the partial fractions.