Problem 1 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{x-12}{x^{2}-4 x}=\frac{A}{x}+\frac{B}{x-4}$$

Step-by-Step Solution

Answer

The values of \(A\) and \(B\) that complete the partial fraction decomposition are \(A=-3\) and \(B=-1\).

Step 1: Setup the equation for manipulating

We'll start by setting up the equation like this: \(x-12 = A(x-4) + Bx\). We can get this equation by multiplying both sides of the initial equation by \(x^{2}-4x\), the common denominator, which will result in this equation.

Step 2: Separate the equation into two parts

Next, we'll set up two equations, setting \(x\) to make either \(A\) or \(B\) drop out. We get \(x=0\) and \(x=4\) which are the roots of the denominator \(x^{2}-4x\). When \(x=0\), the equation becomes \(-12=4A\), and solving for \(A\) gives \(A=-3\). When \(x=4\), the equation becomes \(-4=4B\), and solving for \(B\) gives \(B=-1\).

Step 3: Summary

We found that the values of \(A\) and \(B\) that complete the partial fraction decomposition of the given equation are \(A=-3\) and \(B=-1\) respectively. Therefore, the original equation becomes \(\frac{x-12}{x^{2}-4 x}=\frac{-3}{x}+\frac{-1}{x-4}\).

Key Concepts

Algebra

Algebra is a foundational branch of mathematics that studies symbols and the rules for manipulating these symbols. It's all about finding the unknown values, known as variables, using different mathematical operations. In this exercise, we use algebra to manipulate and simplify the expression

• We start by setting up our equation to apply operations that will reveal the values of unknowns, like constants or variables.
• The key to success in algebra is understanding how to rearrange the equation to isolate these unknown values.
• Techniques such as distributing or factoring might come into play.

When working with partial fraction decomposition, algebra becomes handy. It helps to set equations for breaking down complex fractions into simpler parts.

Rational Functions

Rational functions are fractions consisting of polynomials in both the numerator and the denominator. They're written in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and the denominator \( Q(x) \) is not zero. In this problem, the rational function \( \frac{x-12}{x^{2}-4x} \) is considered.

• To decompose such functions, we look for ways to express them as a sum of simpler rational expressions. This helps in simplifying and solving the equation.
• The factorization of the denominator \( x^2 - 4x \) into \( x(x-4) \) is crucial for setting up the partial fraction decomposition format.
• Each factor gives rise to a separate fraction with a constant numerator that needs to be determined.

Understanding the structure of rational functions allows for effective manipulation and simplification, essential for solving equations that involve these functions.

Equation Solving

Solving equations is the process of finding the values of variables that satisfy the equation. In the step-by-step solution of the original exercise, this principle is applied to find constants \( A \) and \( B \):

• Initially, we transform the equation by clearing fractions, achieved by multiplying both sides by the denominators in the equation.
• This conversion leads us to a simpler polynomial equation \( x-12 = A(x-4) + Bx \).
• The trick is to choose specific values for \( x \) (like the roots of the denominator) to turn parts of the equation into zeros, effectively isolating the unknowns.

Ultimately, understanding how to strategically solve equations is essential. It allows us to move from a complex rational expression to easily solvable linear expressions, unraveling the solution piece by piece. By systematically applying these techniques, finding the values of unknowns becomes a straightforward task.