Question

Find the least common multiple of the expressions. \(2 x, 2 x(x-5)\)

Short answer

The least common multiple of the expressions \(2x\) and \(2x(x-5)\) is \(2x(x-5)\).

Step-by-step solution

Identify the common factor

Firstly, identify the common factor in both expressions. Looking at the given expressions, it can be observed that both expressions can be divided by \(2x\). Therefore, \(2x\) is a common factor.

Identify the remaining term

Secondly, after discovering the common factor, find what is left when this factor is removed from each expression. For the first expression, \(2x\), nothing remains after removing \(2x\) because it comprises the whole expression. For the second expression, \(2x(x-5)\), when the \(2x\) is removed, only \(x-5\) remains.

Construct the least common multiple

Finally, to construct the LCM, all identified factors should be combined. Meaning the LCM is the product of the common factor and the largest remaining terms: \(2x * (x-5)\). Thus, \(2x(x-5)\) is the LCM of \(2x\) and \(2x(x-5)\)

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables (like x or y), and operators (like +, -, *, /). These expressions represent quantities algebraically, and we can manipulate them through various operations. When dealing with algebraic expressions, it's essential to understand terms, coefficients, and like terms. For instance, in the expressions given in the exercise, 2x and 2x(x-5), 2x is a common term and acts like a building block for the expressions.

In our example, the expression 2x(x-5) is also an algebraic expression, where the 2x is multiplied by another expression within the parenthesis, making it a more complex form. Learning to work with algebraic expressions is foundational for further studies in algebra, especially when it comes to finding common multiples or factoring.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler 'factor' expressions that can be multiplied together to get the original polynomial. It's somewhat like finding the original ingredients of a recipe. This process is important for simplifying expressions and solving equations.

In the context of the provided exercise, 2x(x-5) is already presented in factored form. The polynomial x(x-5) has been factored into x and (x-5). Understanding how to factor polynomials can help immensely when trying to find the least common multiple (LCM) or solving for variables. Factoring is a key step in many algebraic processes, so getting comfortable with this concept is crucial for any student of algebra.

LCM in Algebra

The Least Common Multiple (LCM) in algebra is an extension of the LCM concept from arithmetic to algebraic expressions. It represents the smallest expression that is a multiple of two or more algebraic expressions. When calculating the LCM of algebraic expressions, one must consider both the numerical and the variable parts.

Using our exercise as an example, to find the LCM of 2x and 2x(x-5), we identify the common factors and the remaining terms when these common factors are removed. As seen in the step-by-step solution, the common factor is 2x, and after removing 2x from the second expression, we are left with (x-5). By multiplying these together, 2x and (x-5), we get the LCM. Having a strong grasp of this concept is not only essential for simple exercises but also crucial when advancing to more complex algebraic operations involving polynomials.