Problem 8 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &4 x y^{2}\\\ &6 x y^{2}+12 y^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the polynomials \(4xy^2\) and \(6xy^2 + 12y^2\) is \(12xy^2\).

Step 1: Factorize the polynomials

Break down each polynomial into its simplest components: the first polynomial is \(4xy^2 = 2^2 * x * y^2\), and the second polynomial is \(6xy^2 + 12y^2 = 2 * 3 * x * y^2 + 2^2 * 3 * y^2\).

Step 2: Compare the factors

Compare the factored forms of both polynomials and take the highest power of each common factor to get the LCM. Here, the common factors are 2, x, and y^2. The highest power in both polynomials for these factors is 2 for '2', 1 for 'x', and 2 for 'y'.

Step 3: Calculate the LCM

The LCM will be the product of the highest power of all common factors, which is \(2^2 * x * y^2\). However, we must take into account the 3 in the second polynomial since LCM must be a multiple of both polynomials. In this case, the LCM is \(2^2 * 3 * x * y^2\).

Key Concepts

Factoring Polynomials

When it comes to understanding polynomials, one key skill is factoring. Factoring polynomials involves breaking down a polynomial into the product of its simplest components, typically its prime numbers and variables. Think of it like deconstructing a building into its individual bricks. For instance, take the polynomial equation 4xy^2. It can be factored into 2^2 * x * y^2, where 2^2 represents the prime number 2 raised to the second power, x is the variable, and y^2 indicates that the variable y is squared.

To factor polynomials effectively, one should look for the greatest common factor (GCF) among the terms, use the difference of squares, or apply the sum/difference of cubes rules. These methods can simplify complex problems and are the first step in finding polynomial Least Common Multiples (LCM).

Greatest Common Factor

In any arithmetic problem involving fractions or polynomial expressions, identifying the greatest common factor (GCF) is essential. The GCF of two or more polynomials is the highest degree of factor that is common to each term within the polynomial. For example, if you have the polynomials 4xy^2 and 6xy^2 + 12y^2, the GCF is 2xy^2. This is because x and y^2 are common to all terms, and the highest power of 2 that divides into both the coefficients 4 and 6 is 2.

Finding the GCF is essential in performing operations such as simplifying fractions, reducing polynomial expressions, and solving for the LCM of polynomials. With the GCF in hand, you can rewrite and manipulate expressions in a simpler form that can be more easily understood and solved.

Polynomial Arithmetic

Polynomial arithmetic is the bedrock of working with algebraic expressions. It includes operations such as addition, subtraction, multiplication, and finding the LCM or GCD of polynomials. When performing these operations, one must follow the fundamental properties of algebra, like combining like terms and distributing factors.

For LCM calculations, the focus is on multiplication. You identify shared factors between polynomials and then multiply the highest power of these shared factors to get the LCM. Unique factors are also included in the multiplication. The final LCM of two polynomials is the lowest degree polynomial that both original polynomials divide into evenly, making it particularly useful for adding and subtracting polynomial fractions. Remember, LCM involves combining the knowledge of factoring and understanding of the greatest common factor, culminating in the ability to perform polynomial arithmetic effectively.