Problem 84 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{2 x+1}{x-1}-\frac{3}{x+1}+\frac{x}{x-1} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3x^2 + x + 4}{(x-1)(x+1)} \).

Step 1: Identify Common Denominators

First, observe that the denominators of the fractions are \(x-1\) and \(x+1\). The least common denominator (LCD) for all terms is their product, \((x-1)(x+1)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction so they have the LCD:* \(\frac{2x+1}{x-1} = \frac{(2x+1)(x+1)}{(x-1)(x+1)}\)* \(\frac{3}{x+1} = \frac{3(x-1)}{(x-1)(x+1)}\)* \(\frac{x}{x-1} = \frac{x(x+1)}{(x-1)(x+1)}\)

Step 3: Combine All Fractions Over the Common Denominator

Now add and subtract the numerators:\[ \frac{(2x+1)(x+1) - 3(x-1) + x(x+1)}{(x-1)(x+1)} \].

Step 4: Expand and Simplify the Numerator

Expand the expressions in the numerator:- \((2x+1)(x+1) = 2x^2 + 2x + x + 1 = 2x^2 + 3x + 1\)- \(-3(x-1) = -3x + 3\)- \(x(x+1) = x^2 + x\) Combine them: \[2x^2 + 3x + 1 - 3x + 3 + x^2 + x\]. Put them together: \[3x^2 + x + 4\].

Step 5: Write the Final Expression

The simplified expression is then:\[ \frac{3x^2 + x + 4}{(x-1)(x+1)} \].

Key Concepts

Common Denominator

A common denominator is crucial when working with multiple algebraic fractions, especially when adding or subtracting them. Imagine you have fractions with different denominators—in order to combine them into a single fraction, you need a common baseline for each term. This baseline is referred to as the "common denominator."

• In algebra, finding a common denominator involves determining the most comprehensive factor that each term’s denominator can divide without leaving a remainder.
• For example, in the given expression, the denominators are \(x-1\) and \(x+1\). The least common denominator (LCD) to use is \((x-1)(x+1)\).

Having a common denominator allows you to rewrite all terms so they share the same denominator, making them easier to uniformly add or subtract. Once rewritten with the common denominator, the expression can be simplified more easily.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like "regular" fractions, they can be simplified, combined, added, or subtracted. The key is understanding them as fractions made of polynomial terms.

• In simplifying rational expressions, it’s important to keep track of polynomial terms in both the numerator and the denominator.
• Such expressions can be rearranged by finding common denominators, just as we did in the exercise above.

Handling rational expressions involves using algebraic rules for combining fractions. By finding a common denominator, we can easily manage the manipulability of several rational expressions, combining them into a single coherent expression.

Polynomial Simplification

Simplifying polynomial expressions is a critical step in algebra, making complex expressions more manageable and useful. This often involves expanding, simplifying, and combining like terms.

• To simplify a polynomial, you start by expanding any products, as shown in the steps, such as \((2x+1)(x+1)\). This results in a quadratic polynomial when distributed.
• After distributing, terms are collected and like terms are combined to simplify further to a more compact form. In our example, this resulted in \(3x^2 + x + 4\).

The process focuses on reducing complexity without changing the expression’s value. Simplifying polynomials enhances understanding and makes solving or manipulating equations much easier. It is a fundamental skill that aids in tackling more sophisticated algebraic problems.