Problem 25 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{x^{2}+2 x+7}{x(x-1)^{2}} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{x^{2}+2 x+7}{x(x-1)^{2}}\) is \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}\), where A, B, and C values are found by comparing coefficients.

Step 1: Identify Types of Factors in Denominator

Our denominator is \(x(x - 1)^2\). It can be seen that there are two distinct types of factors: \(x\) is a linear factor, and \((x - 1)^2\) is a quadratic factor (i.e., a repeated linear factor). This will direct how we decompose the fraction in the next step.

Step 2: Decompose the Fraction

Generally, a partial fraction decomposition will have the form \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}\). Let's set this decomposition equal to our original fraction: \(\frac{x^{2}+2 x+7}{x(x-1)^{2}} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}\). Now, the next step is to solve for A, B, and C. For this, get rid of the fractions by multiplying the entire equation by \(x(x-1)^2\).

Step 3: Find Constants A, B and C

Upon expanding and comparing coefficients, you will get three equations in terms of A, B and C. Solving this system will give you the values for A, B and C.

Step 4: Write Final Decomposed Fraction

Insert the values of A, B, and C to the decomposed fraction. This fraction is the partial fraction decomposition of the given rational expression.

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but instead of integers, they have polynomials in the numerator, the denominator, or both. The key to working with these types of fractions is understanding that they follow the same rules as the fractions you're already used to. Simplification, addition, subtraction, multiplication, and division all apply, but with a little extra care given the presence of variables.

For example, if you need to add algebraic fractions, you would find a common denominator just like you would with numerical fractions. Simplifying algebraic fractions often involves factoring polynomials and reducing common factors. This is where algebraic skills such as expanding, factoring, and simplifying expressions come into play. Simplification can help in better understanding the structure of the expression, which is especially useful in partial fraction decomposition.

Rational Expressions

Rational expressions are fractions that contain polynomials in both their numerators and denominators. Just as a rational number is the ratio of two integers, a rational expression is the ratio of two polynomials. These expressions can be simplified, multiplied, divided, and even factored just like numerical fractions.

Rational expressions are used in calculus for integration and in algebra for simplifying complex expressions. Partial fraction decomposition is one technique that is particularly useful with rational expressions because it breaks them down into simpler pieces that are easier to work with or integrate.

Linear Factors

Linear factors are the building blocks of polynomials and are expressed in the form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is the variable. When factoring polynomials, we often aim to break them down into their linear factors. This makes it easier to evaluate, graph, or solve equations involving that polynomial.

In the context of partial fraction decomposition, each distinct linear factor in the denominator will correspond to a separate term in the decomposition. For instance, if your denominator includes the factor \(x - 1\), your partial fraction decomposition will include a term \(\frac{A}{x - 1}\). The concept of linear factors is essential in understanding partial fraction decomposition of rational expressions.

Quadratic Factors

Quadratic factors are expressions of the second degree, typically taking the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic factors in the denominator of a rational expression complicate the partial fraction decomposition process, as each factor may result in more than one term in the decomposition.

If the quadratic factor does not repeat, it leads to a single term in the decomposition. But if it does repeat, as in the exercise example \(x - 1)^2\), each instance needs its own term. The coefficient(s) for the quadratic terms may need to be determined by setting up and solving a system of equations, which usually involves equating coefficients of polynomial terms. Understanding how to work with quadratic factors is therefore critical for effectively performing partial fraction decomposition.