Problem 41 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{4 x}{3 x-4}+\frac{8}{3 x^{2}-4 x}+\frac{2}{x}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{2(2x+3)}{3x-4} \).

Step 1: Identify the common denominator

The given expression is \( \frac{4x}{3x-4} + \frac{8}{3x^2-4x} + \frac{2}{x} \). First, factor the denominator of the second fraction: \( 3x^2 - 4x = x(3x - 4) \). The denominators now seen are \( 3x-4, x(3x-4), \text{ and } x \). The common denominator for these fractions is \( x(3x-4) \).

Step 2: Rewrite each term with the common denominator

Convert each fraction to have the common denominator \( x(3x-4) \).- The first term becomes: \( \frac{4x}{3x-4} = \frac{4x \cdot x}{x(3x-4)} = \frac{4x^2}{x(3x-4)} \).- The second term is already: \( \frac{8}{x(3x-4)} \).- The third term becomes: \( \frac{2}{x} = \frac{2 \cdot (3x-4)}{x(3x-4)} = \frac{6x - 8}{x(3x-4)} \).

Step 3: Combine the numerators over the common denominator

With all terms having the common denominator \( x(3x-4) \), we combine the numerators: \[ \frac{4x^2 + 8 + 6x - 8}{x(3x-4)} \].

Step 4: Simplify the numerator

The numerator \( 4x^2 + 8 + 6x - 8 \) simplifies to \( 4x^2 + 6x \).

Step 5: Factor the simplified numerator

Factor \( 4x^2 + 6x \) to get \( 2x(2x + 3) \). Thus, the expression is \( \frac{2x(2x+3)}{x(3x-4)} \).

Step 6: Cancel common factors in the fraction (if any)

The expression \( \frac{2x(2x+3)}{x(3x-4)} \) allows the \( x \) in the numerator and denominator to cancel, resulting in \( \frac{2(2x+3)}{3x-4} \).

Key Concepts

Algebraic Fractions

Algebraic fractions, also known as rational expressions, involve fractions in which the numerator and/or the denominator are polynomials. These types of expressions frequently appear in algebra and require careful handling to simplify and solve. Just like regular fractions, you can add, subtract, multiply, or divide algebraic fractions.
When dealing with algebraic fractions, it's important to:

• Identify the terms in both the numerator and denominator.
• Understand the operations involved, such as addition or multiplication.
• Look for opportunities to simplify by factoring polynomials.

Mastering algebraic fractions involves recognizing patterns and applying basic algebraic rules. Once you're comfortable, these expressions become a powerful tool for solving complex algebraic problems.

Common Denominator

The common denominator in algebraic fractions is crucial for operations like addition or subtraction. Just as with numerical fractions, fractions with differing denominators cannot be directly added. Finding a common denominator helps to bring all the fractions to a uniform base, allowing you to combine them.
To find a common denominator:

• Identify the denominators of all the algebraic fractions involved.
• Factor these denominators fully to reveal common factors.
• Multiply these factors together to find the least common denominator.

This process might seem tricky at first, but practice will make it second nature. In the given exercise, the common denominator is found by recognizing the shared terms among the denominators: \( x(3x-4) \). This serves as the foundational step in combining the fractions correctly.

Simplifying Expressions

Simplifying expressions is often the end goal when working with algebraic fractions. It involves reducing an expression to its simplest form while maintaining its equivalence. Simplification helps in making calculations easier and reveals the core structure of expressions.
To simplify expressions, you generally:

• Factor both the numerator and the denominator whenever possible.
• Cancel out any common factors between the numerator and the denominator.
• Re-write the remaining expression in its simplest form.

In our exercise, simplifying started with factoring the combined numerator \( 4x^2 + 6x \) into \( 2x(2x + 3) \), followed by canceling the common factor \( x \), resulting in a cleaner expression.

Factoring Polynomials

Factoring polynomials is a key skill necessary for simplifying algebraic fractions. It involves expressing a polynomial as a product of its factors, which simplifies operations and makes complex expressions easier to handle. Factoring can be done using different methods depending on the polynomial type.
Common methods of factoring include:

• Factoring out the greatest common factor.
• Using special patterns, such as differences of squares or perfect square trinomials.
• Applying factoring formulas or techniques specific to higher degree polynomials.

In the problem you worked on, \( 3x^2 - 4x \) was factored as \( x(3x-4) \) to help identify the common denominator. Understanding these techniques helps in collaborating with complex algebraic expressions effectively.