Problem 7 - Algebra 2

Question

Find the least common multiple of each pair of polynomials. \(12 x^{2}-6 x-126\) and \(18 x-63\)

Step-by-Step Solution

Answer

The least common multiple of \(12x^{2} - 6x - 126\) and \(18x - 63\) is \( 6 * 9 * 2x * (x-3) * (2x+7)\).

Step 1: Factorizing the polynomials

First, we start by factoring the given polynomials. \(12x^{2} - 6x - 126\) can be factored to \(6(2x^{2} - x - 21)\) and further to \(6(2x+7)(x-3)\). \(18x - 63\) can be factored as \(9(2x - 7)\).

Step 2: Finding the LCM

The least common multiple would be the product of the highest powers of all factors in the two polynomials. So, the LCM is \( 6 * 9 * 2x * (x-3) * (2x+7)\).

Key Concepts

Understanding the Least Common Multiple for Polynomials

The Least Common Multiple (LCM) of polynomials is a crucial concept in algebra. When dealing with polynomial expressions, finding the LCM involves determining the smallest polynomial that each of the given polynomials can divide without leaving a remainder. For numeric values, this involves identifying the largest power of each prime number that appears in the factorization. With polynomials, it includes:

• Identifying each distinct factor in the polynomials.
• Taking the highest power of each factor present in any of the polynomials.

To compute the LCM, once you have the fully factored form of each polynomial, combine these factors by selecting the highest power for each term. This ensures that the LCM contains all necessary factors from both polynomials.

Factoring Polynomials

Factoring polynomials means breaking them down into simpler "building block" expressions that, when multiplied together, would give the original polynomial. For the given polynomials, the strategy is as follows:

• For the polynomial \(12x^2 - 6x - 126\), you begin by finding common elements and grouping. It factors into \(6(2x + 7)(x - 3)\).
• For the polynomial \(18x - 63\), you factor it as \(9(2x - 7)\).

By expressing them as products of simpler polynomials, it becomes easier to manipulate and solve polynomial equations or to find their Least Common Multiple. Factoring is a fundamental skill in algebra, allowing us to simplify expressions and solve equations more effectively.

Exploring Basic Algebra Concepts

Algebra involves working with symbols and letters to represent numbers and quantities in formulas and equations. When exploring polynomials, we use algebraic methods to manipulate and solve these expressions. Key aspects include:

• Understanding variables and coefficients in expressions.
• Recognizing patterns and structures within polynomial expressions, like the distribution of terms.
• Applying rules such as the distributive property to expand expressions.

Solving polynomial problems involves recognizing relationships between expressions and constructing solutions using algebraic operations. Mastering these basic concepts lays the foundation for more advanced algebraic studies.

Working with Polynomial Expressions

Polynomial expressions consist of variables raised to whole number exponents, and coefficients which are the constants multiplying the variables. Understanding polynomials forms the basis for more complex algebraic concepts. In our example:

• Each term in the polynomial \(12x^2 - 6x - 126\) involves a coefficient and a power of \(x\).
• We manipulate these terms through operations like factorization or addition and subtraction of polynomials.

Polynomials are generally arranged in descending order of the power of their variable terms. Working with polynomial expressions helps refine skills in simplification, expansion, and solving algebraic equations, all vital components of mathematics.