Problem 15 - Precalculus: Mathematical for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{12}{x^{2}-9}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{2}{x-3} - \frac{2}{x+3}\).

Step 1: Factor the Denominator

First, identify the denominator of the rational function. The given denominator is \(x^2 - 9\). Notice that \(x^2 - 9\) is a difference of squares, which can be factored as \((x - 3)(x + 3)\).

Step 2: Set Up Partial Fractions

Since \(x^2 - 9 = (x - 3)(x + 3)\), we can express \(\frac{12}{x^2-9}\) as the sum of two partial fractions: \(\frac{A}{x-3} + \frac{B}{x+3}\).

Step 3: Create a Common Denominator

Multiply each term by the common denominator \((x - 3)(x + 3)\) to eliminate the fractions: \[A(x + 3) + B(x - 3) = 12\]

Step 4: Expand and Simplify

Expand the expression on the left side: \[A(x + 3) + B(x - 3) = Ax + 3A + Bx - 3B = 12\]Combine like terms to get: \[(A + B)x + (3A - 3B) = 12\]

Step 5: Solve for A and B

Since there is no \(x\) term on the right side of the equation, set the coefficients of \(x\) equal to zero: \[A + B = 0\]And set the constants equal: \[3A - 3B = 12\]Solve the system of equations: 1. \(A + B = 0\) 2. \(3A - 3B = 12\)From equation 1, we get \(A = -B\).Substitute \(A = -B\) in equation 2:\[3(-B) - 3B = 12 \-3B - 3B = 12 \-6B = 12 \B = -2\]Substitute \(B = -2\) back into \(A = -B\):\[A = 2\]

Step 6: Write the Partial Fraction Decomposition

Substitute \(A = 2\) and \(B = -2\) back into the partial fractions:\[\frac{12}{x^2 - 9} = \frac{2}{x-3} - \frac{2}{x+3}\]

Key Concepts

Rational Function

Rational functions are a combination of two polynomials expressed as a fraction. They're called rational because of the relationship to the word "ratio" due to their fractional form. These functions are in the format \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions and \( Q(x) eq 0 \). Rational functions are interesting because they can have discontinuities and asymptotic behavior due to restrictions when the denominator equals zero.

• Numerator: The divisor of the rational function made up of one or several terms.
• Denominator: The divisor function which must not be zero.

Partial fraction decomposition breaks down complex rational functions into simpler fractions. This technique is especially useful for integration and solving differential equations. Understanding the structure of polynomials involved helps in effectively decomposing the rational function.

Difference of Squares

The difference of squares is a specific type of polynomial that appears very often when decomposing rational functions. It takes the form \( a^2 - b^2 \), and this can be factored into \( (a - b)(a + b) \). Recognizing this pattern is key in simplifying expressions and is essential in partial fraction decomposition.

• Recognize the pattern \( a^2 - b^2 \).
• Factor using \( (a - b)(a + b) \).

For the exercise, the denominator \( x^2 - 9 \) is a classic example of a difference of squares. By identifying and factoring this, we simplify the expression and pave the way to successfully decompose the rational function into partial fractions. This step is crucial in moving forward with any decomposition task.

System of Equations

A system of equations is a set of equations with multiple variables that are solved together. In the context of partial fraction decomposition, once we distribute and simplify terms, we often end up with a system of equations that allows us to solve for unknown coefficients.

• Identify the variables and their respective equations.
• Solve systematically, often through substitution or elimination methods.

In the given solution, after distributing the terms, we equated the coefficients and constants to form a system: \( A + B = 0 \) and \( 3A - 3B = 12 \). Solving this system involved substitution, leading to \( A = 2 \) and \( B = -2 \). Understanding and solving systems of equations is pivotal in partial fraction decomposition, providing the foundational step towards rewriting the fraction in its decomposed form.