Problem 6 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &6 a b^{2}\\\ &18 a b^{3} \end{aligned}$$

Step-by-Step Solution

Answer

Therefore, the LCM of the given polynomials \(6ab^{2}\) and \(18ab^{3}\) is \(18ab^{3}\).

Step 1: Identify the Highest Power of Each Variable

Looking at the provided polynomials, it becomes clear that the power of \(a\) in both polynomials is 1, and thus the highest power of \(a\) is 1. Similarly, the highest power of \(b\) is 3. The variable \(b\) is raised to the 2nd power in the first polynomial and the 3rd power in the second polynomial.

Step 2: Determine the LCM of the Coefficients

The next step is to find the LCM of the coefficients of the variables in the polynomials. The coefficient of \(a\) in both polynomials is 1, and the highest power of this coefficient is 1. The coefficients of the polynomials are 6 and 18, the LCM of which is 18.

Step 3: Combine the Highest Powers of Each Variable with the LCM of the Coefficients

Once the highest power of each variable and the LCM of the coefficients have been determined, these can be combined to calculate the LCM of the polynomials. This results in the polynomial \(18a * b^{3}\).

Key Concepts

Highest Power of Variables

When we talk about the highest power of variables in algebra, we are referring to the exponent of a variable that appears with the largest value in a given term or expression. For example, in the expression x^3y^2 + 2x^4y, the highest power of x is 4 (because of x^4), and the highest power of y is 2 (because of y^2).

This concept is crucial when finding the Least Common Multiple (LCM) of polynomials because the LCM must contain each variable to its highest power that occurs in any of the terms. This assures that each polynomial can divide the LCM without leaving a remainder. In simple terms, just like when working with numbers, if you were to list out multiples of the polynomials, the LCM would be the first one that all the polynomials could fit into perfectly. For instance, if we have polynomials 6ab^2 and 18ab^3, as in the exercise, the variable a appears to the first power in both, and the highest power of b is b^3 in the second polynomial.

LCM of Coefficients

When calculating the LCM of coefficients, it's very similar to finding the LCM of regular numbers. Coefficients are the numerical parts that multiply the variable parts on a term. For instance, in the algebraic terms 6ab^2 and 18ab^3, the coefficients are 6 and 18, respectively. The LCM of these coefficients is the smallest number that both 6 and 18 divide into without leaving a remainder.

To find this, one may list the multiples of 6 (6, 12, 18, 24...) and the multiples of 18 (18, 36, 54...). The first common multiple they share is 18. So the LCM of 6 and 18 is 18. This will form part of the final LCM of the polynomials. It's a crucial step in polynomial operations, especially when adding, subtracting, or comparing polynomials with different coefficients.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division). For example, 2x + 3y - 5 is an algebraic expression.

These expressions are the bread and butter of algebra and are important for representing relationships between different quantities. When dealing with LCM in the context of algebraic expressions, you're typically combining several expressions to form one new expression that contains all elements of the original expressions to their highest powers with the appropriate coefficients. In essence, it's a way to merge expressions into a form that is common to all the expressions you're dealing with, which simplifies addition and subtraction involving polynomials.

Understanding how to handle algebraic expressions is vital for students because these skills apply to a wide range of topics from basic algebra to advanced calculus. It's also key in solving many real-world problems in various fields including science, engineering, and economics.