Problem 13 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &(x-1)(x+2)\\\ &(x-1)(x+3) \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the given polynomials is \((x-1)(x+2)(x+3)\)

Step 1: Identify Common Factors

Look at the given polynomials and identify any common factors. Here, both the polynomials have the factor \(x-1\).

Step 2: Identify Unique Factors

Next, identify the factors that are unique to each polynomial. In this case, one polynomial has the unique factor \(x+2\) and the other polynomial has the unique factor \(x+3\).

Step 3: Find the LCM

The LCM is found by multiplying the common factor and all unique factors together. So, the LCM of these two polynomials is \((x-1)(x+2)(x+3)\)

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. These expressions are formed by combining different terms using addition and subtraction. For example, in the polynomials

• \((x-1)(x+2)\)
• \((x-1)(x+3)\)

The terms can be multiplied together to expand or simplify the expression. Polynomials are fundamental in algebra and used in various operations, making them essential to understand when learning about Least Common Multiple (LCM) and other mathematical concepts.

Recognizing how to break them into factors is crucial when calculating the LCM.

Identifying Common Factors

Common factors are parts of the expressions that appear in each polynomial. For instance, the polynomials

• \((x-1)(x+2)\)
• \((x-1)(x+3)\)

share a common factor, which is \((x-1)\).

Recognizing common elements helps simplify complex calculations by reducing the number of unique elements you need to consider. It's the starting point for finding the LCM of two polynomials by focusing on shared components that appear in both.

The Role of Unique Factors

Unique factors are terms that appear in only one of the polynomials. In our example:

• \((x+2)\) is unique to the first polynomial.
• \((x+3)\) is unique to the second polynomial.

These unique parts play a significant role when finding the LCM, as they must be incorporated into the final expression to ensure that the solution fully represents both polynomials.

By combining these with the common factors, you ensure completeness in your LCM calculation.

Factorization in LCM Calculation

Factorization involves breaking down polynomials into their simplest parts or factors. This strategy is essential for simplifying expressions and finding the LCM.

• First, factor each polynomial to identify common and unique elements.
• Next, multiply the common factor with all unique factors.
• Finally, for our exercise, the LCM is found to be \((x-1)(x+2)(x+3)\).

Factorization not only aids in LCM calculations but also in many other areas of algebra. It allows for more accessible manipulation and understanding of algebraic expressions.