In Exercises 9–42, write the partial fraction decomposition of each rational expression. (2x^2 -18x -12)/x³- 4x

Verified step by step guidance

1. Step 1: Identify the type of partial fraction decomposition needed. Since the degree of the numerator (2) is less than the degree of the denominator (3), we can proceed with partial fraction decomposition.
2. Step 2: Factor the denominator. The denominator is $x^3 - 4x$. Factor out an $x$ to get $x(x^2 - 4)$. Notice that $x^2 - 4$ is a difference of squares, which can be factored further into $(x - 2)(x + 2)$.
3. Step 3: Set up the partial fraction decomposition. Since the denominator factors into $x(x - 2)(x + 2)$, the partial fraction decomposition will be of the form $\frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2}$.
4. Step 4: Clear the fractions by multiplying through by the common denominator $x(x - 2)(x + 2)$. This will give you the equation $2x^2 - 18x - 12 = A(x - 2)(x + 2) + Bx(x + 2) + Cx(x - 2)$.
5. Step 5: Expand and collect like terms on the right-hand side, then equate the coefficients of corresponding powers of $x$ from both sides of the equation to 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 case, the expression (2x^2 - 18x - 12)/(x^3 - 4x) needs to be analyzed to break it down 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 rewrite the given rational expression in a form that is easier to work with, typically involving linear or irreducible quadratic factors in the denominator.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into its constituent factors, which can be multiplied together to yield the original polynomial. This step is essential in partial fraction decomposition, as it allows us to identify the structure of the denominator. For the expression given, factoring the denominator x^3 - 4x will help determine the appropriate form for the partial fractions.