Problem 4 - Precalculus

Question

Match the rational expression \((1-6)\) with the form of the partial-fraction decomposition \((a-f)\). a. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C x+D}{x^{2}+25}\) b. \(\frac{A}{x}+\frac{B x+C}{x^{2}+25}+\frac{D x+E}{\left(x^{2}+25\right)^{2}}\) c. \(\frac{A}{x}+\frac{B x+C}{x^{2}+25}\) d. \(\frac{A}{x}+\frac{B}{x+5}+\frac{C}{x-5}\) e. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C x+D}{x^{2}+25}+\frac{E x+F}{\left(x^{2}+25\right)^{2}}\) f. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+5}+\frac{D}{x-5}\) $$\frac{3 x+2}{x^{2}\left(x^{2}-25\right)}$$

Step-by-Step Solution

Answer

The matching form is option f: \(\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+5}+\frac{D}{x-5}\).

Step 1: Identify Form of Denominator

The given rational expression is \(\frac{3x+2}{x^2(x^2-25)}\). The denominator can be factored as \(x^2(x+5)(x-5)\), which includes the factors \(x^2\), \((x+5)\), and \((x-5)\).

Step 2: Determine Partial Fraction Form

Given the factorization \(x^2(x+5)(x-5)\), the corresponding partial fraction decomposition will involve a term for \(x^2\), a term for \(x+5\), and a term for \(x-5\). This gives the form: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+5} + \frac{D}{x-5}\).

Step 3: Match with Given Options

From the options, the form that matches \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+5} + \frac{D}{x-5}\) is option f. The terms in option f account for the distinct linear factors \((x+5)\) and \((x-5)\) and the repeated linear factor \(x^2\).

Key Concepts

Understanding Rational Expressions

Rational expressions are like fractions. But instead of simple numbers, they involve polynomials in both the numerator and the denominator. These expressions can look intimidating at first, but they follow similar rules as regular fractions, particularly in terms of operations like addition, subtraction, multiplication, and division.
For example,

• If you have a rational expression such as \(\frac{3x+2}{x^2-25}\), the numerator is \(3x+2\), and the denominator is \(x^2-25\).
• The key to working with them is the same as with fractions: make sure the denominator is not zero!

In simplifying or transforming rational expressions, one of the main goals is to either simplify the expression or rewrite it in a different form. This other form could be a partial fraction decomposition, especially useful for integration in calculus.

The Role of Factorization

Factorization is like the detective work of algebra. It involves breaking down algebraic expressions into simpler multiplicative components. Much like how numbers can be factored into prime components, polynomials can be factored into simpler polynomials.
For instance, given the expression \(x^2(x^2-25)\), factorization would transform this into \(x^2(x+5)(x-5)\). We recognize that \((x^2 - 25)\) is a difference of squares, which factors into \((x+5)(x-5)\).
Here's why factorization matters:

• It's essential in simplifying expressions or solving equations.
• It helps to identify opportunities for cancelling out terms.
• It is crucial for the partial fraction decomposition because it helps identify the distinct factors that will dictate the decomposition form.

Mastering factorization opens up a variety of algebraic techniques and offers solutions to many mathematical problems.

Exploring Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. Like their simpler cousins (numeric fractions), they can be simplified, expanded, or broken down to facilitate calculations and solutions in equations.

Partial fraction decomposition is often used with these to break them down into simpler parts. This process is like solving a puzzle where you try to express the fraction as a sum of simpler fractions. For example, taking the expression \(\frac{3x+2}{x^2(x^2-25)}\), once factored into \(x^2(x+5)(x-5)\), can be rewritten as \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+5} + \frac{D}{x-5}\).
This approach:

• Helps in integrating complex algebraic fractions.
• Provides clearer insights into the behavior of the expression across different values.
• Aids in solving equations involving rational expressions.

Algebraic fractions form a crucial part of algebra and calculus, providing essential tools for analysis and problem-solving.