Problem 29 - Algebra A Combined Function

Question

Find the \(L C D\) for each list of rational expressions. $$ \frac{5 x+1}{x^{2}+3 x-4}, \frac{3 x}{x^{2}+2 x-3} $$

Step-by-Step Solution

Answer

The LCD is \((x + 4)(x - 1)(x + 3)\).

Step 1: Factor each denominator

Before finding the Least Common Denominator (LCD), we must factor each denominator. For the first expression, factor \(x^2 + 3x - 4\). The factors are \((x + 4)(x - 1)\). For the second expression, factor \(x^2 + 2x - 3\). The factors are \((x + 3)(x - 1)\).

Step 2: Identify all unique factors

List all unique linear factors that appear in any of the factored expressions. The factors we have are \((x + 4), (x - 1),\) and \((x + 3)\).

Step 3: Construct the LCD

The Least Common Denominator is constructed by taking the product of all unique factors identified. Therefore, the LCD is \((x + 4)(x - 1)(x + 3)\).

Key Concepts

Factoring Polynomial Expressions

Factoring polynomial expressions is a key step in finding the Least Common Denominator (LCD) when dealing with rational expressions. To factor a polynomial expression, you need to write it as a product of simpler polynomials. Consider the quadratic expressions in the original problem: - For the expression \(x^2 + 3x - 4\), the factors are \((x + 4)(x - 1)\).- For the expression \(x^2 + 2x - 3\), the factors are \((x + 3)(x - 1)\).Factoring involves finding numbers or expressions that multiply together to give the original polynomial. The goal is to break down the expression into simpler parts.Notably, understanding how to factor quadratic expressions is fundamental. Use techniques like observation for looking at the constant and coefficient products, or the quadratic formula when expressions are more complex. Once factored, these expressions reveal the building blocks that aid in determining the LCD.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In the context of our exercise, we deal with two rational expressions: - \(\frac{5x+1}{x^2+3x-4}\)- \(\frac{3x}{x^2+2x-3}\)These expressions behave similarly to regular fractions, where operations involve finding a common denominator. The purpose of this is to make the expressions easier to add, subtract, or compare.When working with rational expressions, apply similar rules from arithmetic fractions to manage them. Also, remember always to check for expression simplifications by canceling out common factors between the numerator and the denominator. Though not applicable in the exercise directly, being aware of it is helpful in general practice.

Unique Factors

Identifying all unique factors is crucial when finding the Least Common Denominator for rational expressions. In step 2 of the original solution, every factor from the factored denominators should be considered. Here, we look at each distinct factor that appears in the denominators and list them:- From the first expression, \((x + 4)\) and \((x - 1)\) are factored.- From the second expression, add \((x + 3)\), noting \((x - 1)\) repeats.Unique factors are essentially the building blocks necessary for constructing the LCD. They are crucial, as duplicating factors would unnecessarily complicate the denominator.By understanding this, the LCD is the product of all distinct factors. In simpler terms, you must compile a list of all factors from each denominator, but only include each factor once.