In Exercises 9–42, write the partial fraction decomposition of each rational expression. (6x-11)/(x − 1)²

Verified step by step guidance

1. Identify the form of the rational expression: $\frac{6x-11}{(x-1{)}^2}$.
2. Since the denominator $(x-1{)}^2$ is a repeated linear factor, set up the partial fraction decomposition as $\frac{A}{x-1}+\frac{B}{(x-1{)}^2}$.
3. Multiply both sides by the common denominator $(x-1{)}^2$ to eliminate the fractions: $6x-11=A(x-1)+B$.
4. Expand and simplify the right side: $6x-11=Ax-A+B$.
5. Equate the coefficients of like terms from both sides to form a system of equations and 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 like addition, subtraction, and decomposition. In this case, the expression (6x-11)/(x − 1)² is a rational expression that requires analysis of its components to simplify or decompose.

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 fractions. For the given expression, we will break it down into simpler fractions based on the factors of the denominator, which in this case is (x - 1)².

Polynomial Long Division

Polynomial long division is a process used to divide one polynomial by another, similar to numerical long division. It is essential when the degree of the numerator is greater than or equal to the degree of the denominator. In this problem, if necessary, polynomial long division can help simplify the expression before applying partial fraction decomposition.