Write the partial fraction decomposition of each rational expression. (7x-4)/(x²-x-12)

Verified step by step guidance

1. First, factor the denominator of the rational expression. The denominator is \(x^2 - x - 12\). To factor it, find two numbers that multiply to \(-12\) and add to \(-1\).
2. Rewrite the denominator as the product of two binomials using the numbers found. This will give you something like \((x + a)(x + b)\) where \(a\) and \(b\) are the numbers from the previous step.
3. Set up the partial fraction decomposition. Since the denominator factors into two linear factors, express the rational expression as \(\frac{7x - 4}{(x + a)(x + b)} = \frac{A}{x + a} + \frac{B}{x + b}\), where \(A\) and \(B\) are constants to be determined.
4. Multiply both sides of the equation by the denominator \((x + a)(x + b)\) to clear the fractions. This will give you an equation involving \(7x - 4\) and the expressions \(A(x + b) + B(x + a)\).
5. Expand the right side and collect like terms. Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\) and \(B\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. For example, x^2 - x - 12 factors into (x - 4)(x + 3). This step is essential in partial fraction decomposition to break down the denominator into simpler linear factors.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with linear or irreducible quadratic denominators. It simplifies integration and other operations by breaking complex fractions into manageable parts.

Setting Up and Solving Equations for Coefficients

After expressing the rational function as a sum of partial fractions, you set up equations by equating numerators. Solving these equations for unknown coefficients allows you to find the constants that complete the decomposition.