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

Verified step by step guidance

1. <Step 1: Identify the form of the partial fraction decomposition.> The given rational expression is $\frac{x}{(x-2)(x-3)}$. Since the denominator is a product of two distinct linear factors, the partial fraction decomposition will be of the form $\frac{A}{x-2}+\frac{B}{x-3}$.
2. <Step 2: Set up the equation.> Write the equation $\frac{x}{(x-2)(x-3)}=\frac{A}{x-2}+\frac{B}{x-3}$.
3. <Step 3: Clear the denominators.> Multiply both sides of the equation by $(x-2)(x-3)$ to eliminate the denominators, resulting in $x=A(x-3)+B(x-2)$.
4. <Step 4: Expand and simplify.> Distribute $A$ and $B$ on the right side: $x=Ax-3A+Bx-2B$. Combine like terms to get $x=(A+B)x-(3A+2B)$.
5. <Step 5: Equate coefficients.> Compare the coefficients of like terms on both sides of the equation. For the $x$ terms: $A+B=1$. For the constant terms: $-3A-2B=0$. Solve this system of equations to find the values of $A$ 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 like 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 breaking down the expression based on the factors of the denominator.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential in partial fraction decomposition, as the first step is to factor the denominator completely. Understanding how to factor polynomials helps identify the appropriate form for the partial fractions and ensures accurate decomposition.