In Exercises 9–42, write the partial fraction decomposition of each rational expression. (7x-4)/(x^2-x-12)

Verified step by step guidance

1. Identify the type of the denominator: \(x^2 - x - 12\) is a quadratic expression.
2. Factor the quadratic expression in the denominator: \(x^2 - x - 12 = (x - 4)(x + 3)\).
3. Set up the partial fraction decomposition: \(\frac{7x - 4}{(x - 4)(x + 3)} = \frac{A}{x - 4} + \frac{B}{x + 3}\).
4. Multiply through by the common denominator \((x - 4)(x + 3)\) to clear the fractions: \(7x - 4 = A(x + 3) + B(x - 4)\).
5. Expand and collect like terms on the right side, then equate coefficients to solve for \(A\) and \(B\).

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 such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is often necessary for integration or solving equations.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful when integrating rational expressions, as it allows for easier manipulation and integration of each term. The process involves factoring the denominator and expressing the original fraction in terms of its simpler components.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler polynomial factors that, when multiplied together, yield the original polynomial. This is essential in partial fraction decomposition, as the first step involves factoring the denominator to identify the appropriate form for the decomposition, which can include linear and irreducible quadratic factors.