Problem 21 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{6 x-11}{(x-1)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{6x-11}{(x-1)^{2}}\) is \(\frac{-5}{x-1} - 5\).

Step 1: Write the General Form of the Partial Fraction

First, express the rational expression in the general form of the partial fraction decomposition. Since the denominator \((x-1)^2\) is a perfect square, the expected general form of the partial fraction is \( \frac{6x - 11}{(x - 1)^2} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} \) where A and B are constants that we'll find next.

Step 2: Clear Fraction by Multiplying Through by the Common Denominator

In order to find the constants A and B, multiply the general form by the common denominator \((x - 1)^2\). Doing this gives: \(6x - 11 = A(x - 1) + B\). This equation is true for all values of x, so we can choose convenient values for x to find A and B.

Step 3: Solve for A and B

Choose x = 1 to solve for B, as this will make A(0) go away. Substituting x = 1 into the equation in step 2, we get: \(6*1 - 11 = B\), or \(B = -5\). Next, substitute x = 0 to solve for A, which makes A(-1) - B = 0. Therefore, A = B, or A = -5.

Key Concepts

Rational Expression

A rational expression is similar to a fraction, but instead of integers or real numbers in its numerator and denominator, it contains algebraic expressions. In mathematics, these are often encountered when dealing with equations or functions that involve division of polynomials.

As an example, in the exercise provided, \( \frac{6x-11}{(x-1)^2} \) is a rational expression where the numerator is the polynomial 6x-11 and the denominator is the polynomial \(x-1)^2\). To work with these expressions effectively, especially when it comes to integration or finding limits, it's often helpful to break them down into simpler parts using partial fraction decomposition, which provides a more convenient form for further manipulation.

Common Denominator

The concept of a common denominator is pivotal when dealing with fractions and rational expressions. In basic arithmetic, a common denominator between two fractions is a shared multiple of their original denominators; it allows for the addition or subtraction of these fractions. Similarly, in algebra, when dealing with rational expressions that have different polynomials in the denominator, we often seek a common denominator to combine or compare them.

During the partial fraction decomposition process, identifying the common denominator allows us to clear the fraction and simplify the expression into a set of algebraic equations that can be solved for specific values. This step is crucial in the provided exercise, as multiplying the entire equation by the common denominator \( (x - 1)^2 \) makes it possible to isolate and solve for the unknown constants in the decomposed form.

Algebraic Equations

An algebraic equation is a mathematical statement that includes an equals sign and algebraic expressions on either side. These equations are the foundation for solving a wide range of problems in mathematics. The goal is to find the value of an unknown variable that makes the equation true. This can involve a variety of techniques such as simplifying expressions, factoring, and manipulating the equation to isolate the variable.

In the context of partial fraction decomposition, after clearing the fractions and setting up our equation \(6x - 11 = A(x - 1) + B\), we turn our attention to solving for the unknown constants A and B. By strategically choosing certain values for x, such as those that simplify the equation, we can find the values of A and B. This is akin to solving a system of linear equations, which may be more familiar from earlier algebra classes.