Problem 46 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{3 x}{x-3}-\frac{x+4}{x+2} $$

Step-by-Step Solution

Answer

\[\frac{2(x - 1)(x + 6)}{(x - 3)(x + 2)}\]

Step 1: Identify Least Common Denominator

The denominators are \(x-3\) and \(x+2\). The least common denominator (LCD) for fractions is the smallest number both denominators can divide into evenly. In this case, the LCD is the product of the two denominators (since they are dissimilar): \(LCD = (x-3)(x+2)\).

Step 2: Convert the Fractions to Have the Same Denominator

Multiply the numerator and the denominator of the first fraction by \(x+2\) and the numerator and the denominator of the second fraction by \(x-3\) to have a common denominator. After performing the operation, the expression becomes \[\frac{3x(x + 2)}{(x - 3)(x + 2)} - \frac{(x + 4)(x - 3)}{(x - 3)(x + 2)}\]

Step 3: Perform Addition or Subtraction Operation

Since we now have common denominators, we can do the subtraction in the numerators: \[\frac{3x^2 + 6x - x^2 + 4x - 12}{(x - 3)(x + 2)}\] This simplifies to: \[\frac{2x^2 + 10x - 12}{(x - 3)(x + 2)}\]

Step 4: Simplify the Result

The final step is to simplify the expression. Factoring the numerator yields: \[\frac{2(x - 1)(x + 6)}{(x - 3)(x + 2)}\] There are no common factors to cancel between the numerator and the denominator, so this is our final answer.

Key Concepts

Understanding the Least Common Denominator

When adding or subtracting rational expressions, the first thing to find is the least common denominator (LCD). This is similar to finding a common denominator in simpler fractions, but with one key difference — we're dealing with polynomials instead of numbers. The LCD is the smallest expression that each denominator can divide into evenly, helping us combine the fractions.

To identify the LCD, look at each denominator of the rational expressions involved. If they contain different terms, multiply the distinct denominators together. For example, if you have denominators like

• \( (x - 3) \)
• \( (x + 2) \)

Your least common denominator will be the product of these two: \( (x - 3)(x + 2) \). By using this LCD, each fraction can be rewritten to share the same denominator, which is essential for straightforward addition or subtraction.

The Role of Factoring Polynomials

Factoring polynomials is a fundamental part of simplifying rational expressions. When you have expressions in the numerator or denominator, factoring can often simplify complex terms into more manageable pieces. This process involves breaking down a polynomial into a product of simpler polynomials, much like breaking a number into its prime factors.

For instance, suppose our expression is \( 2x^2 + 10x - 12 \). We need to see if this can be factored further. Here are steps to factor polynomials:

• Look for a common factor for all terms first, such as a number or variable.
• Apply techniques like grouping or the quadratic formula if necessary.
• In our example, factor it as: \( 2(x - 1)(x + 6) \).

Factoring helps in simplifying and reducing expressions, especially when looking for potential terms that could cancel with the denominator. This is crucial for making the expression as simple as possible, but only after combining the expressions.

Simplifying Rational Expressions for Clarity

After factoring, the next step is simplifying rational expressions as much as possible. Simplification makes expressions easier to process and interpret, much like reducing fractions to their simplest form. It typically involves canceling out any common factors between the numerator and the denominator.

To simplify, follow these general steps:

• Make sure every term in the numerator and denominator is fully factored.
• Check if there are any common factors.
• Cancel out these common factors. In our example with \( \frac{2(x - 1)(x + 6)}{(x - 3)(x + 2)} \), we found no common factors and thus the expression stays the same.

The goal of simplifying is to reach a form that is easily readable and interpretable, confirming that the solution is as straightforward as possible. This step ensures precision and understanding, highlighting the underlying structure of the mathematical expression.