Problem 23 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &x^{2}-2 x-24\\\ &x^{2}-36 \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the given polynomials \(x^{2}-2x-24\) and \(x^{2}-36\) is \((x-6)(x+4)(x+6)\)

Step 1: Factorize the given expressions

First, factorize \(x^{2}-2x-24\) into \((x-6)(x+4)\) and \(x^{2}-36\) into \((x+6)(x-6)\)

Step 2: Identify the common factors

After factoring, identify the common factors from among the given expressions. In this case, the common factor is \((x-6)\)

Step 3: Calculate the LCM

The LCM is calculated by taking the highest power of all the factors from both expressions. The LCM of the given expressions is therefore \((x-6)(x+4)(x+6)\)

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down numbers into primes, but instead, you deal with algebraic expressions. It involves expressing a polynomial as a product of simpler polynomials, known as factors. For example, from the exercise, the polynomial \(x^2 - 2x - 24\) is changed into its factors \((x - 6)(x + 4)\). Factoring allows you to simplify algebraic expressions and solve polynomial equations.

Here are some key points about factoring:

• Common Patterns: Look for patterns such as the difference of squares, perfect square trinomials, and sum or difference of cubes.
• Decomposition: Break down the middle term to find two numbers that multiply to the product of the leading coefficient and the constant term.
• Trial and Error: Sometimes, using different numbers and terms until the factors come together correctly can be useful.
• Factoring by Grouping: Involves collecting terms into groups and factoring each group.

By recognizing these patterns and techniques, you can effectively factor any polynomial, simplifying the subsequent steps in finding the LCM.

Common Factors

Identifying common factors in polynomials is a crucial step when finding the Least Common Multiple (LCM). Common factors refer to the elements or terms that appear in each set or expression. Finding these shared elements helps in simplifying problems, especially in tasks like finding the LCM.

Here's how you can identify common factors:

• Factor Each Polynomial: Write down the polynomials as a product of their factors. For instance, from the given polynomials \((x - 6)(x + 4)\) and \((x + 6)(x - 6)\), you can clearly see the common factor \((x - 6)\).
• Check Repeated Elements: Look for factors that appear more than once across the expressions. These are your common factors.
• Use the Common Factor for Simplification: In many mathematical operations, especially finding the LCM, the presence of common factors helps reduce redundancy.

Ultimately, recognizing common factors allows for efficient problem solving and streamlines processes such as combining fractions or simplifying expressions.

Highest Power of Factors

When calculating the LCM of two polynomials, it's essential to take the highest power of all the factors present in either polynomial. This ensures that the LCM indeed represents a common multiple that each original polynomial can divide without remainder.

Let's break down this concept:

• List All Factors: First, determine the factors for each polynomial. For example, the factors for the polynomials in our exercise are \((x-6), (x+4),\) and \((x+6)\).
• Identify the Highest Power: Among these, choose the highest power for each distinct factor. Usually, when polynomials don't have repeated powers, you write each factor once.
• Construct the LCM: Multiply these highest power factors together. The result for our polynomials is \((x-6)(x+4)(x+6)\), ensuring it encompasses all original polynomial factors at least once.

By considering the highest power of each factor, the LCM ensures coverage of all terms, making it a versatile expression for various algebraic operations.