Question

Find the least common denominator of the expressions.\(\frac{x-1}{x+2}, \frac{8}{x^{2}-x-6}, \frac{x}{x-3}\)

Short answer

The least common denominator (LCD) of the expressions is \((x+2)(x-3)\)

Step-by-step solution

Factorize the Denominators

Factorise the denominators to its simplest form. So now we have: \(\frac{x-1}{x+2}, \frac{8}{(x-3)(x+2)}, \frac{x}{x-3}\)

Identify the Factors

Identify the factors in each denominator. In this case we have \(x+2\) and \(x-3\)

Find the LCD

Find the least common denominator by multiplying all the distinct factors present in the denominators. In this case as \(x+2\) and \(x-3\) both are distinct, the least common denominator(LCD) is \((x+2)(x-3)\)

Key concepts

Factorization

Factorization is a fundamental aspect of algebra that deals with breaking down numbers, algebraic expressions, or polynomials into their simplest building blocks. Think of it like dismantling a complex structure into its basic components, making it easier to work with or understand. Imagine you have a number like 12; factorization would involve expressing it as 2 x 2 x 3, which are its prime factors.

When we apply this concept to algebraic expressions, particularly polynomial denominators, we need to identify factors that are common to each term. This makes it possible to simplify expressions and solve equations more efficiently. For instance, in our problem, the denominator \(x^2 - x - 6\) can be factorized into \((x-3)(x+2)\). Each factor represents a simpler expression that, when multiplied together, yields the original polynomial. Understanding factorization is especially crucial when finding the least common denominator (LCD) because it requires you to identify and use the simplest form of each denominator.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, they do not have an equality sign; they are rather expressions of relationships that can be simplified and manipulated algebraically. In the context of our problem, \(\frac{x-1}{x+2}\), \(\frac{8}{x^2 - x - 6}\), and \(\frac{x}{x-3}\) are all algebraic expressions.

The main goal when working with algebraic expressions in this context is to express them in a form that allows for easy identification of common elements, such as the least common denominator. As expressions become more complex with variables and multiple terms, a clear understanding of how to handle and simplify these expressions is key. Simplifying algebraic fractions to find the LCD involves ensuring the denominators are broken down to their simplest form, which then enables you to combine the fractions if needed.

Polynomial Denominators

Polynomial denominators are the bottom parts of fractions where the numerator is divided by a polynomial, e.g., a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Finding the least common denominator (LCD) that involves polynomial denominators can be quite challenging as it requires a good grasp of factorization. The LCD is essentially the smallest polynomial that each of the denominators in a set of algebraic fractions can divide into without leaving a remainder.

In our sample problem, the denominators upon factorization reveal common factors. Detecting these common factors is vital because they determine the LCD. As soon as you have factorized the denominators, you can identify the unique factors that make up the LCD. For the given set of fractions, the LCD is the product of all unique factors, which in this case is \((x+2)(x-3)\). Identifying the LCD is essential for adding, subtracting, or comparing algebraic fractions because it allows the fractions to be expressed with a common base, facilitating these operations.