Problem 5 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &8 x^{3} y\\\ &12 x y^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the polynomials \(8x^{3}y\) and \(12xy^{2}\) is \(2^{3} * 3 * x^{3} * y^{2}\)

Step 1: Factorize the Polynomials

First, express each polynomial as a product of prime factors. For the polynomial \(8x^{3}y\), the factorization is \(2^{3} x^{3}y^1\). Similarly, for the polynomial \(12xy^{2}\), the factorization is \(2^{2}3x^1y^{2}\)

Step 2: Identify and Select the Highest Powers of each Factor

Now, identify the highest powers of each prime factor from the factorization of both polynomials. The highest power of 2 is \(2^{3}\), the highest power of 3 is 3, the highest power of x is \(x^{3}\) and the highest power of y is \(y^{2}\)

Step 3: Form the LCM

Finally, form the LCM by multiplying together the highest powers of each prime factor. The LCM is then, \(2^{3} * 3 * x^{3} * y^{2}\)

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking apart a number to see what it’s made of, just with variables instead of numerals. It's an essential skill when finding the Least Common Multiple (LCM) of polynomial expressions. Here's how it works: each polynomial is expressed in its most broken-down form. This is similar to how we break down numbers into their prime factors, but with polynomials, we are looking at the expressions inside variables.

• First, look at the coefficients and factor them into their prime components. For example, the coefficient 8 in the polynomial \(8x^3y\) can be factored into \(2^3\), and for 12 in \(12xy^2\), it's \(2^2 \times 3\).
• Next, consider the powers of the variables. In expressions like \(x^3\) and \(y^2\), these indicate how many times a variable is multiplied by itself. You treat these like prime factors too.

Breaking down a polynomial in this manner sets the stage for easily finding the highest powers of each factor, making the process less daunting.

Highest Powers

Identifying the highest powers of each factor is crucial when calculating the LCM of polynomials. After you've factored the polynomials, the next step is to look at both sets of factors and choose the largest ones for each component.For example, after factoring the polynomials \(8x^3y\) and \(12xy^2\):

• The highest power of 2 occurs in \(2^3\), since it's greater than \(2^2\).
• The factor 3 only appears in one polynomial, so you take \(3\).
• For variable \(x\), the highest power is \(x^3\).
• For variable \(y\), the largest exponent is \(y^2\).

By selecting the highest powers from your factored forms, you ensure that the LCM can divide each polynomial without any remainder. This step ensures inclusivity of all possible components.

Prime Factorization

Prime factorization involves breaking down numbers or terms into their most basic building blocks, which are prime numbers in simple arithmetic. With polynomial expressions, you extend this concept to include variables as well.When factoring polynomials, identify and break them down to their unique prime number components as well as variable factors raised to their respective powers.

• The number 8 in \(8x^3y\) is broken down to \(2^3\), emphasizing that 2 is prime.
• The number 12 in \(12xy^2\) becomes \(2^2 \times 3\), both primes.
• Variables are considered as prime factors with their powers reflecting their magnitude in the expression.

By applying prime factorization to polynomials, you lay down a systematic path for identifying and working with highest powers to easily determine the LCM.

Polynomial Expressions

Polynomial expressions are combinations of variables and coefficients constructed through addition, subtraction, and multiplication. They can range from very simple, such as \(x\), to more complex forms like \(8x^3y\).Polynomials follow certain rules that make operations predictable:

• They often consist of constants (like 8 or 12), variables (like \(x\) and \(y\)), and exponents which tell you what power the variable is raised to.
• Understanding polynomials is foundational for algebra since they form the building blocks of algebraic equations.
• Knowing how to manipulate and factor polynomial expressions is critical in algebra to solve more complex problems like finding LCM or solving equations.

Getting familiar with polynomial expressions allows you not only to find the LCM but also aids in learning to manage diverse problems in mathematics with confidence.