Problem 34 - Precalculus Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{-2 x^{2}+5 x-1}{x^{4}-2 x^{3}+2 x-1}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{-2}{x^2 - x + 1} + \frac{7x + 1}{(x^2 - x + 1)^2} \).

Step 1: Understand the Equation

We are given the rational function \( \frac{-2x^2 + 5x - 1}{x^4 - 2x^3 + 2x - 1} \). Our task is to express this as a sum of simpler fractions (partial fractions). The denominator is a polynomial of degree 4, indicating multiple roots.

Step 2: Factor the Denominator

The denominator \( x^4 - 2x^3 + 2x - 1 \) needs to be factored into simpler terms. This polynomial can be challenging, but after testing some values and using synthetic division or factoring by grouping, it can be factored into \((x^2 - x + 1)^2\).

Step 3: Set Up the Partial Fraction Form

Since the denominator factors to \((x^2 - x + 1)^2\), the partial fraction decomposition will take the form: \[\frac{A x + B}{x^2 - x + 1} + \frac{C x + D}{(x^2 - x + 1)^2}\]where \(A, B, C,\) and \(D\) are constants to be determined.

Step 4: Combine and Clear Fractions

Multiply both sides by the original denominator \((x^2 - x + 1)^2\) to eliminate the fractions. Equate:\[-2x^2 + 5x - 1 = (A x + B)(x^2 - x + 1) + (C x + D)\].

Step 5: Expand and Collect Terms

Expand the right-hand side:\[(A x + B)(x^2 - x + 1) = A x^3 - A x^2 + A x + B x^2 - B x + B\]Combine this with the terms \( C x + D \). Then collect terms into a polynomial of the form \( E x^3 + F x^2 + G x + H\).

Step 6: Match Coefficients

Match the coefficients of the polynomial obtained from the expansion with those in \(-2x^2 + 5x - 1\):1. Coefficient of \(x^3\): \(A = 0\)2. Coefficient of \(x^2\): \(-A + B = -2\)3. Coefficient of \(x\): \(A - B + C = 5\)4. Constant term: \(B + D = -1\)

Step 7: Solve the System of Equations

From the system of equations:1. \(A = 0\)2. \(-A + B = -2 \Rightarrow B = -2\)3. \(A - B + C = 5\Rightarrow -2 + C = 5\Rightarrow C = 7\)4. \(B + D = -1\Rightarrow -2 + D = -1\Rightarrow D = 1\)Thus, \(A = 0, B = -2, C = 7, D = 1\).

Step 8: Write the Final Decomposition

Substitute the constants back into the partial fractions:\[\frac{-2}{x^2 - x + 1} + \frac{7x + 1}{(x^2 - x + 1)^2}\]This is the required partial fraction decomposition.

Key Concepts

Rational Function

A rational function is essentially a fraction where both the numerator and the denominator consist of polynomial expressions. This means that you have a polynomial divided by another polynomial. Understanding rational functions is important because they can often be seen as more complex algebraic expressions.

To work with rational functions effectively, you often need to simplify them, identify asymptotic behavior, or decompose them into simpler fractions. Such decomposition is known as partial fraction decomposition, which helps in performing operations like integration and further simplifying complex expressions. In this type of task, we take a given rational function and express it as a sum or combination of simpler, individual fractions.

This methodology not only helps in mathematical problem-solving but also enhances understanding, making it a foundational concept in higher-level algebra and calculus.

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex polynomial into a product of simpler polynomials. This is a crucial step in partial fraction decomposition. When you have a polynomial in the denominator of a rational function, factoring it allows you to express the function as a sum of simpler fractions.

In the provided problem, the denominator is a fourth-degree polynomial, which can often lead to multiple roots. To factor this polynomial, techniques such as synthetic division or factoring by grouping can be employed.

After successfully factoring, you can write the polynomial as a product of its factors, making it easier to perform further operations, such as setting up partial fractions for decomposition. It's essential to practice factoring polynomials regularly, as mastery of this skill is crucial in solving many algebraic problems.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, acting similarly to arithmetic long division you might perform with numbers. This process is often necessary when you want to simplify rational expressions or when the degree of the numerator is higher than the degree of the denominator.

In the context of partial fractions, polynomial long division might be used prior to decomposition if the polynomial in the numerator has a degree equal to or greater than that in the denominator. This step ensures that the rational function is proper, making the partial fraction decomposition valid and simpler to perform.

Understanding polynomial long division involves knowing how to distribute terms, align degrees, and systematically reduce the polynomial step by step. This skill is not only essential for rational functions but is a versatile tool in algebra.

Synthetic Division

Synthetic division is another approach to dividing polynomials which is often quicker and requires less computation than full polynomial long division. It's especially useful when dividing by a linear factor of the form \(x - c\).

In the problem exercise, synthetic division helps to test potential roots and factor the polynomial in the denominator. By using synthetic division, you can efficiently handle polynomials, particularly when they appear complex or cumbersome to factor manually.

This technique involves a streamlined method of performing polynomial division, which simplifies the division process, minimizing errors and calculations. Synthetic division is particularly useful in partial fraction decomposition as it assists in factorizing the denominator polynomial and finding its roots more straightforwardly.