Problem 96 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \(\frac{2x + 3}{x^2 + x}\).

Step 1: Identify the Complex Fraction Structure

First, observe that the complex fraction consists of two parts: a numerator \(2+\frac{1}{x^2 - 1}\) and a denominator \(1+\frac{1}{x-1}\). Identify the inner fractions so we can simplify them separately.

Step 2: Simplify the Inner Fractions

Notice that \(x^2 - 1\) can be factored as \((x-1)(x+1)\). This means the fraction \(\frac{1}{x^2 - 1}\) can be written as \(\frac{1}{(x-1)(x+1)}\). Similarly, observe that the denominator has a fraction \(\frac{1}{x-1}\), which is already simplified.

Step 3: Find a Common Denominator for the Numerator

In the numerator, \(2 + \frac{1}{(x-1)(x+1)}\) requires a common denominator. Represent \(2\) as \(\frac{2(x-1)(x+1)}{(x-1)(x+1)}\) so both terms have the same denominator: \(\frac{2(x-1)(x+1) + 1}{(x-1)(x+1)}\).

Step 4: Simplify the Denominator

For the denominator \(1 + \frac{1}{x-1}\), find a common denominator of \(x-1\). Rewrite \(1\) as \(\frac{x-1}{x-1}\), giving: \(\frac{x-1 + 1}{x-1}\). Simplify to \(\frac{x}{x-1}\).

Step 5: Simplify the Complex Fraction

Replace the complex fraction with: \[ \frac{\frac{2(x-1)(x+1) + 1}{(x-1)(x+1)}}{\frac{x}{x-1}} \]. Invert the denominator and multiply: \[ \frac{2(x-1)(x+1) + 1}{(x-1)(x+1)} \times \frac{x-1}{x} \].

Step 6: Cancel Common Terms

Cancel \((x-1)\) from the numerator and denominator: \[ \frac{2(x+1) + \frac{1}{x-1}}{x(x+1)} \]. It becomes \(\frac{2(x+1) + 1}{x(x+1)} = \frac{2x + 2 + 1}{x(x+1)} = \frac{2x + 3}{x^2 + x}\).

Step 7: Verify the Simplified Expression

Ensure that no further simplification can be done. Since \(\frac{2x + 3}{x^2 + x}\) has no further common factors, this form is the simplest version of the original complex fraction.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In math, they help represent and solve problems involving unknown quantities. For example, in the complex fraction \[ \frac{2+\frac{1}{x^{2}-1}}{1+\frac{1}{x-1}}, \]we see numbers and the variable \(x\).These expressions often involve:

• constants (fixed numbers, like 2): these don't change in value and serve as a baseline for calculations.
• variables (like \(x\): these are placeholders for values that we may not know right away.

Understanding algebraic expressions is crucial, as they form the foundation of many algebra topics. It’s essential to recognize how different parts come together. When dealing with complex fractions, we often break down these expressions into simpler forms to better understand and solve them.

Simplification Techniques

Simplification techniques in algebra are methods used to make expressions easier to work with. Here are some useful simplification steps with explanations:

• **Identifying Components:** Recognize the main parts of the complex fraction, such as the numerator and denominator, as seen in \(\frac{2+\frac{1}{x^{2}-1}}{1+\frac{1}{x-1}}\).Articulate each sub-part for better clarity.
• **Finding Common Denominators:** To simplify fractions, ensure that all terms have a common denominator, helping in adding or subtracting fractions. For instance, in the numerator, convert \(2\) into a fraction with \((x-1)(x+1)\) as the denominator.
• **Inverting and Multiplying:** When dividing by a fraction, such as in complex fractions, invert the denominator and multiply by the numerator. This step is a key simplification technique that streamlines calculations.

These techniques make simplifying complex fractions like ours more approachable, breaking down larger expressions into manageable parts.

Factoring Polynomials

Factoring polynomials involves rewriting an expression as a product of simpler polynomials. This is useful when simplifying complex algebraic expressions, including fractions. In our exercise:

• **Recognize Polynomials:** Identify the polynomial parts, such as \(x^2 - 1\), that can be simplified further.
• **Apply Factoring Techniques:** Factor \(x^2 - 1\) to \((x-1)(x+1),\) a common first step in simplification. Recognizing this formula is crucial, as it belongs to a special category known as "difference of squares."
• **Simplifying Further:** Once factored, these expressions often cancel out with other parts of the fraction, thus simplifying it overall.

Understanding how to factor polynomials correctly allows us to handle complex algebraic equations with greater ease. It is especially helpful in fraction work, where reducing expressions leads to simpler solutions.