Problem 43 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{1+\frac{6}{x}+\frac{8}{x^{2}}}{1+\frac{1}{x}-\frac{12}{x^{2}}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Problem

We are given the complex fraction \( \frac{1+\frac{6}{x}+\frac{8}{x^{2}}}{1+\frac{1}{x}-\frac{12}{x^{2}}} \). Our goal is to simplify it into a more manageable form.

Step 2: Find the Least Common Denominator (LCD)

The denominators within both the numerator and the denominator of the complex fraction are \(1\), \(x\), and \(x^2\). The least common denominator of these is \(x^2\).

Step 3: Rewrite Each Term with the LCD

Convert each term to have the common denominator \(x^2\). For the numerator: \(1 = \frac{x^2}{x^2}\), \(\frac{6}{x} = \frac{6x}{x^2}\), and \(\frac{8}{x^2} = \frac{8}{x^2}\). This makes the numerator \(\frac{x^2 + 6x + 8}{x^2}\). For the denominator: \(1 = \frac{x^2}{x^2}\), \(\frac{1}{x} = \frac{x}{x^2}\), and \(-\frac{12}{x^2} = -\frac{12}{x^2}\). This makes the denominator \(\frac{x^2 + x - 12}{x^2}\).

Step 4: Divide Numerator by Denominator

The expression can now be simplified by dividing the rewritten numerator by the rewritten denominator: \[ \frac{\frac{x^2 + 6x + 8}{x^2}}{\frac{x^2 + x - 12}{x^2}} = \frac{x^2 + 6x + 8}{x^2 + x - 12} \].

Step 5: Factor Both Polynomials

Factor the numerator \(x^2 + 6x + 8\). It factors to \((x + 2)(x + 4)\). Factor the denominator \(x^2 + x - 12\). It factors to \((x + 4)(x - 3)\).

Step 6: Cancel Common Factors

The common factor \((x + 4)\) in both the numerator and the denominator can be canceled: \[ \frac{(x + 2)(x + 4)}{(x + 4)(x - 3)} = \frac{x + 2}{x - 3} \].

Step 7: Provide the Simplified Expression

The simplified form of the original complex fraction is \( \frac{x + 2}{x - 3} \).

Key Concepts

Least Common Denominator

To simplify complex fractions, finding the Least Common Denominator (LCD) is a crucial step. The LCD is the smallest expression that can simplify all denominators in the given fractions. Here, the complex fraction is \(\frac{1+\frac{6}{x}+\frac{8}{x^{2}}}{1+\frac{1}{x}-\frac{12}{x^{2}}}\), containing multiple denominators: 1, \(x\), and \(x^2\).
The goal is to eliminate these by converting each fraction to have the same denominator.
To find the LCD:

• Evaluate the highest power for each variable present in the denominators.
• Consider any constant denominators.

Here, since the denominators are \(1\), \(x\), and \(x^2\), the LCD becomes \(x^2\). This allows us to rewrite each fraction with \(x^2\) as the common denominator, facilitating simplification.

Factoring Polynomials

Factoring polynomials is an important technique used after rewriting the fractions with a common denominator. In our example, you have polynomial expressions like \(x^2 + 6x + 8\) and \(x^2 + x - 12\).
Factoring means expressing a polynomial as a product of its simplest factors. For example:

• The term \(x^2 + 6x + 8\) can be factored into \((x + 2)(x + 4)\).
• Similarly, \(x^2 + x - 12\) factors into \((x + 4)(x - 3)\).

To do this, look for two numbers that multiply to the constant term but add up to the coefficient of the linear term. Breaking down such expressions makes it easier to identify common terms in the numerator and denominator, setting the stage for further simplification in the next steps.

Simplification

Simplifying a complex fraction often involves several stages, including finding the Least Common Denominator and factoring polynomials, as covered earlier. The final stage is simplification, where you cancel out common factors. Starting with:\[\frac{(x + 2)(x + 4)}{(x + 4)(x - 3)}\]One of the shared factors in both the numerator and the denominator is \((x + 4)\).
By canceling this out, you're left with:\[\frac{x + 2}{x - 3}\]This is much simpler than the original complex fraction and is the desired result. Simplification significantly reduces complexity by removing redundancies.
Always remember, when canceling factors, ensure that these terms are non-zero to avoid any undefined expressions. This step is crucial in ensuring the fraction maintains its integrity while being reduced to its simplest form.