Question

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{3}{x^{2}-9}=\frac{A}{x-3}+\frac{B}{x+3}$$

Short answer

The values that complete the partial fraction decomposition are \(A = 0.5\) and \(B = -0.5\).

Step-by-step solution

Equation Setup

First, multiply \(A\) and \(B\) by their denominators and set the expression equal to the numerator of the given fraction. This will give \(3 = A(x + 3) + B(x - 3)\).

Find Value for A

One way to deduce the value of \(A\) is to zero out \(B\) by choosing \(x = 3\). Substituting this into the equation gives \(A = \frac{3}{6} = 0.5\).

Find Value for B

To find the value for \(B\), zero out \(A\) by choosing \(x = -3\). Substituting into the equation gives \(B = \frac{3}{-6} = -0.5\).

Key concepts

Calculus

Understanding calculus begins with grasping foundational algebraic techniques, such as partial fraction decomposition. This mathematical method is particularly useful in calculus when integrating complex rational expressions. Integration, a core concept of calculus, is the process of finding the function that describes the accumulated area under a curve. When faced with integrating a fraction that has a polynomial in the denominator, such as \(\frac{3}{x^2-9}\), breaking it into simpler parts (partial fractions) makes the integration possible or considerably easier.

It's like taking a complex machinery apart to understand and work with each individual component; once you understand each simple motion, putting it together to make sense of the overall function becomes almost natural. Similarly, in calculus, mastering partial fractions equips you with a tool to tackle integrations that involve more complex algebraic fractions.

Algebraic Fractions

Algebraic fractions are simply fractions with polynomials in the numerator, the denominator, or both. The expression \(\frac{3}{x^2-9}\) is a classic example of an algebraic fraction. These fractions can be intimidating due to the presence of variables, but they are manipulated using the same principles that apply to numerical fractions.

Common Denominator Approach

Think of when you add fractions; you find a common denominator. Algebraic fractions often require you to work backward by starting with a common denominator (in this case, \(x^2-9\)) and then breaking it down into simpler expressions. This technique simplifies many algebraic processes, including simplifying complex fractions, solving equations, and performing calculus operations.

Rational Expressions

Rational expressions are ratios of polynomials, much like fractions are ratios of numbers. In partial fraction decomposition, you are breaking down a complicated rational expression into simpler ones that are easier to work with.

Understanding the behavior of these expressions is critical, especially when functions approach certain values that could make the denominator zero (a.k.a. vertical asymptotes). Through the process of decomposition, like the one seen in the exercise, we disassemble the original rational expression into components that reveal more about the function's behavior and allow for further analysis or simplification.

Equation Solving

Solving equations is the bread and butter of algebra. In the partial fraction decomposition exercise, we solve for variables \(A\) and \(B\) within the constraints of the equation. This involves a series of strategic moves to isolate the variables, such as zeroing out terms or substituting values for \(x\) that simplify the expression.

Strategies for Isolation

For instance, by choosing \(x = 3\), we cleverly remove \(B\) from the equation, making it possible to solve for \(A\). A similar process is used to find \(B\). These strategies draw from our fundamental equation-solving skills and demonstrate the importance of logical steps and critical thinking in algebra. Manipulating and solving equations not only enables you to find specific values but also deepens your understanding of the relationships between different parts of an expression.