Problem 17 - Algebra and Trigonometry

Question

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{12}{x^{2}-9} $$

Step-by-Step Solution

Answer

\(\frac{12}{x^2 - 9} = \frac{2}{x - 3} - \frac{2}{x + 3}\)

Step 1: Factoring the Denominator

First, notice that the denominator can be factored. The expression in the denominator is a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \]This decomposition will help us to express the rational function as a sum of partial fractions.

Step 2: Setting Up the Partial Fractions

With the denominator factored, set up the partial fraction decomposition:\[\frac{12}{x^2 - 9} = \frac{A}{x - 3} + \frac{B}{x + 3}\]where \( A \) and \( B \) are constants that need to be determined.

Step 3: Clear the Denominator

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

Step 4: Expand and Collect Like Terms

Expand the right-hand side of the equation:\[12 = Ax + 3A + Bx - 3B\]Combine like terms:\[12 = (A + B)x + (3A - 3B)\]

Step 5: Set Up and Solve Equations for Constants

For the equation to hold for all \(x\), the coefficients of like terms must be equal. This gives us a system of equations:1. \( A + B = 0 \)2. \( 3A - 3B = 12 \)Solve these equations: - From equation 1, \( A = -B \) - Substitute into equation 2: \( 3(-B) - 3B = 12 \) which simplifies to \(-6B = 12\) and \( B = -2 \). - Substitute back to find \( A: A = 2 \).

Step 6: Write the Partial Fraction Decomposition

Now that we have \( A = 2 \) and \( B = -2 \), substitute back into the partial fractions:\[\frac{12}{x^2 - 9} = \frac{2}{x - 3} - \frac{2}{x + 3}\]This is the partial fraction decomposition of the given rational function.

Key Concepts

Rational Function

A rational function is a type of function that can be expressed as a fraction where both the numerator and the denominator are polynomials. Here, we are focused on the rational function \[\frac{12}{x^2 - 9}\]Understanding rational functions is essential for various mathematical analyses, including integration and asymptotic behavior. In this context, we decompose the function into simpler parts, making it easier to understand or compute further. Notably, a rational function may have:

• Vertical asymptotes, where the denominator equals zero causing the function to approach infinity.
• Horizontal or slant asymptotes, determined by the degrees of the numerator and denominator.

Partial fraction decomposition is a tool often used in calculus for integrating rational functions like the one above.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler terms or factors that, when multiplied together, will yield the original polynomial. In our exercise, the denominator of the rational function \[x^2 - 9\]can be factored using a special rule called the difference of squares. This rule states that \[a^2 - b^2 = (a - b)(a + b)\]Applying this rule:\[x^2 - 9 = (x - 3)(x + 3)\]Factoring simplifies the expression and is crucial in setting up the partial fraction decomposition. It allows us to break down complex expressions into easier, more manageable parts.

System of Equations

A system of equations is a set of simultaneous equations, which we solve to find common solutions to each equation. In the context of partial fraction decomposition, these systems arise while determining unknown constants in the decomposed fractions.After expressing the function in terms of partial fractions, we cleared the fraction and solved:\[12 = A(x + 3) + B(x - 3)\]Expanding and collecting like terms, we obtain: \[12 = (A + B)x + (3A - 3B)\]This results in a system:

• \(A + B = 0\)
• \(3A - 3B = 12\)

Solving these, we use substitution or elimination methods to find:\(A = 2\) and \(B = -2\). Such systems ensure all like terms across equations equate, enabling correct decomposition.

Difference of Squares

The difference of squares is a specific technique in polynomial factorization that can simplify expressions significantly. It is based on the identity:\[a^2 - b^2 = (a - b)(a + b)\]This identity applies perfectly when a polynomial looks like the one found in our rational function denominator \(x^2 - 9\). By recognizing it as a difference of squares, we wrote it as:\[(x - 3)(x + 3)\]The process is pivotal in the partial fraction decomposition strategy, making it easier to break down and manipulate expressions. Understanding the difference of squares allows us to identify hidden structures within polynomials and is an important skill in algebra, calculus, and beyond. It turns complex mathematical expressions into simpler, more tractable forms.