Problem 23 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{4 x}{x^{2}-4} \cdot \frac{x+2}{16 x}$$

Step-by-Step Solution

Answer

\(\frac{1}{4(x-2)}\)

Step 1: Identify and Factor Denominators

Start by identifying the denominators of each fraction. The denominators are \( x^2 - 4 \) and \( 16x \). Factor the denominator \( x^2 - 4 \) as \((x - 2)(x + 2)\) since it is a difference of squares.

Step 2: Write the Full Expression

The original multiplication expression can be rewritten with factored denominators as: \( \frac{4x}{(x-2)(x+2)} \cdot \frac{x+2}{16x} \).

Step 3: Simplify by Cancelling Common Terms

Check the numerators and denominators for any common factors. The terms \( x + 2 \) and \( 4x \) in the numerator and denominator can be cancelled because they appear in both the numerator and the denominator. After cancelling, you're left with \( \frac{1}{(x-2) \cdot 4} \) or \( \frac{1}{4(x-2)} \).

Step 4: Final Simplification

The final simplified form of the expression is \( \frac{1}{4(x-2)} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental step in simplifying rational expressions, especially when dealing with problems that require multiplication or division. A polynomial is factored into simpler terms, called factors, which multiply to give back the original polynomial. There are different methods to factor polynomials, such as:

• Factoring out the greatest common factor (GCF): This involves finding the largest factor that all terms share and factoring it out.
• Factoring by grouping: This is used when a polynomial has four or more terms, where terms are grouped to find and factor out the GCF in pairs.
• Difference of squares: This method uses the identity \(a^2 - b^2 = (a-b)(a+b)\). It applies when a polynomial can be expressed as the difference of two perfect squares.

Factoring simplifies complexity and prepares the expression for further simplification, like cancelling common factors. In our problem, \(x^2 - 4\) was factored using the difference of squares method, resulting in \((x-2)(x+2)\), which sets the stage for further simplification.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, making them easier to work with. This process includes:

• Factorization of numerators and denominators: Begin by completely factoring both the numerator and the denominator of the fraction.
• Cancel common factors: Identify and cancel out common factors that appear in both the numerator and the denominator. This step is crucial in simplifying the expression.

For example, in our task, after factoring, \(4x\) and \(x+2\) were identified as common factors and cancelled out, simplifying the original complex fraction. Resultantly, the expression becomes \(\frac{1}{4(x-2)}\). This final fraction is easier to interpret and work with for any subsequent operations.

Difference of Squares

The difference of squares is a specific factoring technique applicable to any expression that can be written as \(a^2 - b^2\). This identity states that such an expression can be factored into \((a-b)(a+b)\). This method is extremely useful for factoring specific quadratic expressions.Here's how it works:

• Identify the two perfect squares: The original polynomial \(x^2 - 4\) can be seen as \((x)^2 - (2)^2\).
• Apply the difference of squares formula: Using \((a-b)(a+b)\), we express \(x^2 - 4\) as \((x-2)(x+2)\).

This technique not only aids in simplifying the expression but often reveals factors that can be cancelled, which is what features in our exercise solution process. Recognizing and using the difference of squares effectively is a skill that makes simplifying complex expressions dramatically easier.