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

Verified step by step guidance

1. Factor the denominator $x^3+x^2-2x$ into $x(x+2)(x-1)$.
2. Set up the partial fraction decomposition: $\frac{4x^2-3x-4}{x(x+2)(x-1)}=\frac{A}{x}+\frac{B}{x+2}+\frac{C}{x-1}$.
3. Multiply through by the common denominator $x(x+2)(x-1)$ to clear the fractions: $4x^2-3x-4=A(x+2)(x-1)+Bx(x-1)+Cx(x+2)$.
4. Expand and simplify the right-hand side to form a single polynomial expression.
5. Equate the coefficients of corresponding powers of $x$ from both sides to form a system of equations and solve for $A$, $B$, and $C$.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this context, the expression (4x^2 - 3x - 4)/(x^3 + x^2 - 2x) is a rational expression that needs to be decomposed into simpler fractions.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, based on the factors of the denominator.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can be linear or irreducible quadratic expressions. This step is essential in partial fraction decomposition, as it allows us to identify the form of the simpler fractions. For the denominator x^3 + x^2 - 2x, factoring helps determine the appropriate partial fractions to use in the decomposition process.