Question

In Exercises 9-16, find the least common multiple of the expressions. \(3 x, 3(x-2)\)

Short answer

Therefore, the least common multiple (LCM) of the given expressions is \(3x\).

Step-by-step solution

Simplify the expressions

Firstly, simplify the mathematical expressions. The first expression is \(3x\), which is in its simplest form. The second expression simplifies to \(3x-6\). So, you have the two expressions \(3x\) and \(3x-6\)

Factorize the expressions

Factorizing the expressions helps to compare them. The factorized form of the first expression, \(3x\), is \(3 \cdot x\). The second expression factorizes to \(3 \cdot (x-2)\)

Compare and find the LCM

The LCM is the product of 3, x and (x-2). This simplifies to \(3 \cdot x \cdot (x-2)\). However, x and (x-2) are not the same term, so they can't be used together in the LCM. The largest term that satisfies both expressions is 3x.

Key concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in algebra. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). For example, in the exercise provided, we have two expressions: 3x and 3(x-2). These expressions represent certain quantities in terms of variables. When dealing with algebraic expressions, it's crucial to know how to simplify and manipulate them to solve problems such as finding the least common multiple (LCM).

Manipulating algebraic expressions can involve

• Combining like terms,
• Factoring,
• Expanding, and
• Simplifying complex fractions.

In the provided exercise, the expressions are already simplified, but understanding how to perform these operations is key to working with algebraic expressions effectively.

Factorization

Factorization is a process of breaking down a complex expression into a product of simpler factors. It's like taking a number apart to see which other numbers multiply together to make it. In algebra, this often means breaking down an algebraic expression into a product of its simplest building blocks or 'factors'. For instance, the expression 3x is already factored as it is the product of 3 and x. However, the expression 3(x-2) is written in a factored form where 3 is multiplied by the entire expression (x-2).

Effective factorization can:

• Make it easier to simplify expressions,
• Help solve equations and inequalities,
• Allow us to find the least common multiple of algebraic expressions.

While the concept might seem daunting, factorization merely requires practice and familiarization with various algebraic identities and factoring techniques such as the greatest common factor, trinomials, and difference of squares.

LCM in Algebra

The Least Common Multiple (LCM) in algebra extends the concept from numerical LCM to algebraic expressions. When finding the LCM of algebraic expressions, the goal is to determine the smallest expression that all the original expressions can divide into without any remainder. In the case of 3x and 3(x-2), their LCM must have factors that cover both expressions.

To find the LCM:

• Factor all expressions completely.
• For each distinct factor, take the highest power of that factor that appears in any of the expressions.
• Multiply these chosen factors together for the LCM.

In our exercise, the factor 3 is common, and x and (x-2) are two separate factors. The LCM, therefore, becomes the product of the three, written as 3x(x-2). This represents the smallest expression that both 3x and 3(x-2) can divide into completely.