Write the partial fraction decomposition of each rational expression. 4/(2x² -5x -3)

Verified step by step guidance

1. Start by factoring the denominator of the rational expression \(\frac{4}{2x^2 - 5x - 3}\). To do this, look for two numbers that multiply to \(2 \times (-3) = -6\) and add to \(-5\).
2. Rewrite the middle term \(-5x\) using the two numbers found in the previous step, then factor by grouping. This will give you the factored form of the denominator as a product of two binomials.
3. Set up the partial fraction decomposition by expressing \(\frac{4}{(\text{factored form})}\) as a sum of two fractions with unknown constants in the numerators. For example, if the denominator factors as \((ax + b)(cx + d)\), write it as \(\frac{A}{ax + b} + \frac{B}{cx + d}\).
4. Multiply both sides of the equation by the original denominator to clear the fractions. This will give you an equation involving polynomials where the numerators on the right side are combined over a common denominator.
5. Expand and collect like terms on the right side, then equate the coefficients of corresponding powers of \(x\) from both sides to form a system of equations. Solve this system to find the values of \(A\) and \(B\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with linear or quadratic denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. This step is essential in partial fraction decomposition to break down the denominator into simpler factors for easier fraction separation.

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 needed to complete the decomposition.