Problem 24 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{1}{4 x^{2}-9}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{1}{4 x^{2}-9}\) is \(\frac{1/3}{{2x-3}} + \frac{1/3}{2x+3}\)

Step 1: Identify the Form of the Rational Expression

The given expression can be considered as a rational expression, in the form, \(\frac{A}{(2x-3)} + \frac{B}{(2x+3)}\) where \(A\) and \(B\) are constants and need to be found.

Step 2: Create Equations to Find Constants

To find the constants \(A\) and \(B\), equate the initial rational expression and the separated form: \[\frac{1}{4x^2-9} = \frac{A}{2x-3} + \frac{B}{2x+3}\]Now, multiply through by the common denominator \(4x^2-9 = (2x-3)(2x+3)\) to remove the denominators. This gives the equation: \[1 = A(2x+3) + B(2x-3)\]This equation can be solved for two unknowns \(A\) and \(B\) by choosing convenient values for \(x\).

Step 3: Solve for Constants A and B

Setting \(x = \frac{3}{2}\) in the equation, the term \((2x+3)\) will become 0, hence eliminating \(A\). This yields \(A = \frac{1}{3}\).Now, by letting \(x = -\frac{3}{2}\), the term \((2x-3)\) will become 0, hence eliminating \(B\). This gives \(B=\frac{1}{3}\).

Step 4: Writing the Final Partial Decomposition

By substituting the values of \(A\) and \(B\), the partial fraction decomposition of the original expression is thus:\[\frac{1}{4x^2-9} = \frac{1/3}{2x-3} + \frac{1/3}{2x+3}\]

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Much like fractions that contain integers, rational expressions can be simplified, added, subtracted, multiplied, and divided. An important aspect of working with rational expressions is to factor the polynomial in the denominator when possible, as it allows for the simplification of the expression and often is a prerequisite step for partial fraction decomposition.

In the provided exercise, the expression \(\frac{1}{4x^2-9}\) is considered a rational expression because it has a polynomial in the denominator, which factors as the difference of squares \( (2x-3)(2x+3) \). Partial fraction decomposition is used to break down complex rational expressions into simpler fractions that are easier to integrate or differentiate if that is the end goal.

Algebraic Fractions

Rational expressions that involve variables, as in the given problem, are often referred to as algebraic fractions. The process of breaking these down into simpler parts or 'partial fractions' can be seen as the reverse of finding a common denominator to combine fractions. Algebraic fractions are useful in various fields of mathematics, particularly in calculus, where they make finding antiderivatives simpler.

The exercise required a partial fraction decomposition, which separates the complex expression into a sum of simpler algebraic fractions. To successfully decompose the expression \(\frac{1}{4x^2-9}\), it’s essential to factor the denominator first and then determine the numerators for the resulting fractions, keeping in mind that these numerators are constants because the denominator's factors are linear terms.

Solving Equations

Solving equations is a fundamental part of algebra, which involves finding the values of the unknowns that make the equation true. In the case of partial fraction decomposition, the original rational expression is set equal to a sum of fractions with unknown numerators. These numerators are the 'constants' we aim to solve for.

Through carefully selected values of \(x\), which make each denominator factor equal to zero separately, we can 'turn off' terms in the equation, isolating the constants one at a time. This way of 'turning off' terms to solve for unknowns greatly simplifies the process. Once the constants are found, they are substituted back into the partial fractions, providing the decomposed version of the original algebraic fraction.