Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{2}-x-21}{\left(x^{2}+4\right)(2 x-1)} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{-2x + 1}{x^2 + 4} + \frac{5}{2x - 1}\).

Step 1: Identify the Denominator Factors

First, observe that the denominator is already factored as \((x^2 + 4)(2x - 1)\). Since \(x^2 + 4\) is a quadratic factor and \(2x - 1\) is a linear factor, we will have separate terms for each of these factors in the partial fraction decomposition.

Step 2: Set Up the Partial Fraction Decomposition

Write the partial fraction decomposition structure based on the factors of the denominator:\[\frac{x^{2}-x-21}{(x^2 + 4)(2x - 1)} = \frac{Ax + B}{x^2 + 4} + \frac{C}{2x - 1}\]Here, \(Ax + B\) corresponds to the quadratic factor and \(C\) corresponds to the linear factor.

Step 3: Clear the Denominator

Multiply both sides by the common denominator \((x^2 + 4)(2x - 1)\) to clear the fractions:\[x^2 - x - 21 = (Ax + B)(2x - 1) + C(x^2 + 4)\]

Step 4: Expand & Simplify

Expand the right-hand side of the equation:\[(Ax + B)(2x - 1) = 2Ax^2 - Ax + 2Bx - B\]and\[C(x^2 + 4) = Cx^2 + 4C\]Add these expressions together:\[2Ax^2 - Ax + 2Bx - B + Cx^2 + 4C = (2A + C)x^2 + (-A + 2B)x + (-B + 4C)\]

Step 5: Equate Coefficients

Equate the coefficients of the left-hand side \(x^2 - x - 21\) with the expanded right-hand side to find \(A\), \(B\), and \(C\):- For \(x^2\): \(2A + C = 1\)- For \(x\): \(-A + 2B = -1\)- Constant term: \(-B + 4C = -21\)

Step 6: Solve the System of Equations

Solve the system of equations:1. \(2A + C = 1\)2. \(-A + 2B = -1\)3. \(-B + 4C = -21\)By solving these, you find \(A = -2\), \(B = 1\), \(C = 5\).

Step 7: Write the Decomposition

Substitute \(A\), \(B\), and \(C\) back into the partial fraction decomposition:\[\frac{x^2 - x - 21}{(x^2 + 4)(2x - 1)} = \frac{-2x + 1}{x^2 + 4} + \frac{5}{2x - 1}\]

Key Concepts

Rational Expressions

Rational expressions are fractions involving polynomials in the numerator and the denominator. In math, a polynomial is an algebraic expression composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, the expression \(\frac{x^2 - x - 21}{(x^2 + 4)(2x - 1)}\) is a rational expression.Key points about rational expressions:

• The numerator and denominator must both be polynomials.
• These expressions can be simplified, added, subtracted, multiplied, and divided much like regular fractions, provided that the denominator is not zero.
• Simplifying rational expressions often involves factoring polynomials and reducing the fraction like you would with numerical fractions.

Understanding rational expressions forms the basis for operations like partial fraction decomposition, where a complex rational expression is separated into simpler fractions. This simplification makes integration and solving equations much easier.

Algebraic Fractions

Algebraic fractions are akin to rational expressions, consisting of a numerator and a denominator that are both polynomials. These expressions allow us to perform algebraic operations on fractions just as we do with numerical fractions. For instance, in the problem \(\frac{x^2 - x - 21}{(x^2 + 4)(2x - 1)}\), the idea is to break down the algebraic fraction into simpler parts that are more easily dealt with.When dealing with algebraic fractions:

• Ensure the algebraic fraction is in its simplest form. This may involve factoring the numerator and the denominator and reducing common factors.
• Perform operations such as addition, subtraction, multiplication, and division on algebraic fractions by following the basic rules of fractions, while considering the polynomials involved.
• Keep an eye on the restrictions that the denominator imposes to avoid undefined operations, like division by zero.

Once simplified, algebraic fractions can be decomposed using methods like partial fraction decomposition, making them easier to handle for solving equations or integrating expressions.

System of Equations

A system of equations consists of multiple equations that are solved together to find common variable values. In partial fraction decomposition, once you set up your fractions according to the factors in the denominator, you're left with a system of equations to determine the coefficients of your fractions.Let's look at the system of equations from our problem:

• For \(x^2\): \(2A + C = 1\)
• For \(x\): \(-A + 2B = -1\)
• Constant term: \(-B + 4C = -21\)

To solve this system:

• You need to find values for \(A\), \(B\), and \(C\) that satisfy all the equations simultaneously.
• You'll often use methods like substitution or elimination. In this case, solving these equations carefully gives us \(A = -2\), \(B = 1\), \(C = 5\).
• Once these variables are identified, substitute them back into the decomposition structure to finalize the decomposed expression.

Mastering systems of equations is crucial not only for partial fraction decomposition but also for numerous applications in algebra and beyond.