In Exercises 9–42, write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

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1. Step 1: Identify the form of the partial fraction decomposition. Since the denominator is a product of two distinct linear factors, \((x - 9)(x + 2)\), the partial fraction decomposition will have the form \(\frac{A}{x - 9} + \frac{B}{x + 2}\).
2. Step 2: Set up the equation. Write the original rational expression as equal to the sum of the partial fractions: \(\frac{3x + 50}{(x - 9)(x + 2)} = \frac{A}{x - 9} + \frac{B}{x + 2}\).
3. Step 3: Clear the denominators by multiplying through by \((x - 9)(x + 2)\) to eliminate the fractions: \(3x + 50 = A(x + 2) + B(x - 9)\).
4. Step 4: Expand the right side of the equation: \(3x + 50 = Ax + 2A + Bx - 9B\).
5. Step 5: Combine like terms and equate coefficients. This gives you a system of equations to solve for \(A\) and \(B\): \(A + B = 3\) and \(2A - 9B = 50\).

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 the focus of this question.

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 expressions or simplifying complex algebraic fractions, allowing for easier manipulation and analysis of the expression.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler polynomial factors that, when multiplied together, yield the original polynomial. This is essential for partial fraction decomposition, as the first step often requires factoring the denominator to identify the appropriate form for the decomposition.