Problem 14 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &(2 x-1)(x+4)\\\ &(2 x+1)(x+4) \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the two polynomials \( (2 x-1)(x+4) \) and \( (2 x+1)(x+4) \) is \( (2x-1)(2x+1)(x+4) \)

Step 1: Identify the Factors

Identify the unique factors in the polynomials. In this case, there are three unique factors across the two polynomials: \(2x-1\), \(2x+1\), and \(x+4\)

Step 2: Combine All the Factors for the LCM

The LCM of the polynomials is obtained by multiplying the highest power occurring in the polynomials of each factor. In this case, all the factors have power 1 across the two polynomials. Thus, we multiply all the unique factors together. That is, LCM will be: \(LCM = (2x-1)(2x+1)(x+4)\)

Step 3: Simplify

In this case, since the LCM was directly determined from multiplication of factors, there is no need for further simplification.

Key Concepts

Factoring Polynomials

To solve the problem of finding the least common multiple (LCM) of polynomials, first, we need to factor each polynomial into simpler parts. In essence, factoring is breaking down a polynomial into simpler polynomials that multiply together to give the original polynomial. This allows us to see the building blocks of the polynomial. Why do we need this? Because to find the LCM, we need to see all the pieces involved. Think of it like looking at the ingredients of a recipe. We want to make sure we have every necessary component. In the given exercise, each polynomial is already factored for us:

• The first polynomial is expressed as \((2x - 1)(x + 4)\)
• The second polynomial is \((2x + 1)(x + 4)\)

Each of these expressions, like \((2x - 1)\), \((2x + 1)\), and \((x + 4)\) are factors, meaning they are the smaller polynomials that fully multiply back to the original polynomial.

Unique Polynomial Factors

When working with the LCM of polynomials, finding the unique polynomial factors is crucial. Unique factors help us ensure that our LCM is as simple and efficient as possible.In polynomials, unique factors are those elements which appear at least once in any of the given polynomials. Unlike common factors seen in integers where we usually talk about the greatest common factor, in the case of polynomials, we’re focusing on what each polynomial contributes to the whole. Looking at the given exercise, the unique factors across the two polynomials are:

• \((2x - 1)\)
• \((2x + 1)\)
• \((x + 4)\)

This means that for these polynomials, the LCM must include each of these factors, ensuring that when multiplied, all the original polynomials can be constructed. This concept guarantees that we’re creating the smallest possible complete product that accounts for every factor, without duplicating unnecessarily.

Polynomial Multiplication

Polynomial multiplication is essential when finding the LCM because it combines all our unique factors into a single product. Although this may sound complicated, the process is straightforward once you understand each step.To achieve this multiplication, we bundle all identified unique factors:

• \((2x - 1)\)
• \((2x + 1)\)
• \((x + 4)\)

We multiply these together just like regular numbers. The LCM for the given polynomials is hence \((2x - 1)(2x + 1)(x + 4)\). Since each factor is taken only to the highest power it appears in any polynomial, we don’t have repeated terms, and each polynomial's structure is preserved in the LCM.
Multiplying these expressions follows the principles of distributing each component over the others to acquire each component of the final polynomial. The resulting LCM isn't simplified further, as indicated in the original problem, meaning this product is our final answer.