Problem 73 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expanding the second expression

The second expression is \(2(x-5)\). To expand, each term inside the bracket should be multiplied by 2. Doing this yields \(2x-10\).

Step 2: Finding LCM of the expressions

Now the task is to find the LCM of \(2x\) and \(2x-10\). The LCM of two algebraic expressions is the product of the highest powers of all the factors, present in the numbers. So the LCM of the 2 expressions is \(2x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like x or y), and operators, such as addition or multiplication. These expressions are fundamental in algebra because they help us describe general rules and patterns. When working with algebraic expressions, it's essential to understand what each part of the expression represents.
Variables in these expressions serve as placeholders that can take different values, depending on the situation. For example, in the expression \(2x\), "2" is a constant multiplier and "x" is a variable.
Overall, understanding algebraic expressions is crucial because they form the basis of more complex topics, including the operations covered next.

Expansion of Expressions

The expansion of expressions refers to the process of removing grouping symbols, like parentheses, by distributing the multiplying factor to each term inside. This is useful because it makes the expression simpler and easier to work with, especially when finding the least common multiple (LCM).
In the provided exercise, we expanded the expression \(2(x-5)\). To do this, we multiplied each term in the parenthesis by 2.

• First, multiply 2 by \(x\): \(2 \times x = 2x\)
• Then, multiply 2 by \(-5\): \(2 \times -5 = -10\)

After expanding, we obtained \(2x - 10\), which allows us to compare it with another expression for further operations.

Multiplication of Polynomials

Multiplication of polynomials involves multiplying each term of one polynomial by every term of another. This technique is critical for finding factors or simplifying expressions in algebra.
Even though the exercise involves multiplication in the context of expanding an expression, the underlying principle can be seen in polynomial multiplication as well.
When trying to find the least common multiple (LCM) of expressions like \(2x\) and \(2x - 10\), we look at the highest powers of all factors. In simpler terms, we focus on combining like terms and factors to form a singular expression that serves as a common multiple.

• For \(2x\), the factor is simply 2 times x.
• For \(2x - 10\), the factors can be seen as the expanded form after considering the distributive multiplication.

By understanding these multiplications, you can easily determine the LCM of such algebraic expressions, simplifying and solving more complex equations efficiently.