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Chapter 7: Problem 48

Find the LCM. $$4(x-1), 8(x-1)$$

Short Answer

Expert verified
The LCM is \(8(x-1)\).

Step by step solution

01

Identify the common factor

Both expressions share the common factor \(x-1\).
02

Find the LCM of coefficients

Identify the coefficients of the expressions. In this case, \(4\) and \(8\). The least common multiple (LCM) of these coefficients is \(8\).
03

Form the LCM

Combine the LCM of the coefficients with the common factor to get the final LCM. Hence, the LCM is \(8(x-1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
When looking at algebraic expressions like these, identifying a common factor can make our life much easier. But what is a common factor? Simply put, a common factor is a term that appears in each expression you are comparing. In our case, both expressions, $$4(x-1)$$ and $$8(x-1)$$, have the term \(x-1\) multiplied by a coefficient. This shared part, \(x-1\), is our common factor. Recognizing this common piece helps us to focus on the other parts that vary between the expressions, simplifying our problem-solving process.
Coefficients
Coefficients play a crucial role in algebra, especially when determining the Least Common Multiple (LCM). A coefficient is the numerical part of a term that is multiplied by the variable. In the given expressions, $$4(x-1)$$ and $$8(x-1)$$, the coefficients are 4 and 8, respectively. These numbers alone, without the variables, are what we're interested in when finding the LCM of the coefficients. Understanding how to handle and operate on coefficients is essential for solving algebraic problems efficiently.
LCM Calculation
Finding the LCM of numbers or terms is a common task in both arithmetic and algebra. LCM stands for Least Common Multiple, which is the smallest number that is evenly divisible by both coefficients involved. In our example, the coefficients are 4 and 8. The smallest number that both 4 and 8 can divide without leaving a remainder is 8, making 8 the LCM of these two coefficients. Combining this with our earlier identified common factor, we get the full LCM of the algebraic expressions.
Algebraic Expression
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Expressions like $$4(x-1)$$ and $$8(x-1)$$ are typical examples. Here, \(x-1\) acts as a constant term combined with a variable, and it is the part of the expression that doesn't change as we find the LCM. The coefficients, 4 and 8, determine the numerical part. Algebraic expressions must be carefully managed to solve for LCMs, common factors, or any other operations correctly. Practice with these can help solidify your understanding and build a strong foundation in algebra.

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