Finding the LCM of 23x 4 - 9x^2 and 2 - 3x: A Comprehensive Guide

Understanding the Least Common Multiple (LCM) is essential for various algebraic operations. In this article, we will explore how to find the LCM of the expressions 23x 4 - 9x2 and 2 - 3x, explaining the steps and methodologies used in the process.

Introduction to Algebraic Expressions

Algebraic expressions such as 23x 4 - 9x2 and 2 - 3x require a thorough understanding of the principles of algebra. These expressions represent polynomial functions, which are used in a wide range of applications, from mathematics to engineering and physics.

Factorization of the Expressions

Before finding the LCM, we first need to factorize the given expressions. Let's begin with 23x 4 - 9x2.

Factorizing 23x 4 - 9x2

The expression 23x 4 - 9x2 can be written as:

23x 4 - 9x2 23x2 - 9x2

Note that both terms have a common factor of x2. Factoring out x2 gives:

23x2 - 9x2 x2(23 - 9x)

Factorizing 2 - 3x

The expression 2 - 3x is already in its simplest form and cannot be further factorized. Therefore, it remains as:

2 - 3x

Determining the LCM

Once the expressions are factorized, we can determine the LCM by identifying the highest power of each factor present in the expressions.

Identifying the Factors

The factors in the expression 23x 4 - 9x2 are x2 and (23 - 9x).

The factors in the expression 2 - 3x are simply (2 - 3x).

The LCM is the product of the highest power of each factor found in both expressions.

Calculating the LCM

The LCM of x2 and (2 - 3x) is:

LCM x2 × (2 - 3x)

Therefore, the LCM of 23x 4 - 9x2 and 2 - 3x is x2(2 - 3x).

Conclusion

In conclusion, finding the LCM of algebraic expressions involves a systematic approach. By factorizing the given expressions and identifying the highest power of each factor, we can determine the LCM accurately. In this case, the LCM of 23x 4 - 9x2 and 2 - 3x is x2(2 - 3x).

Additional Resources

For further study on algebraic expressions and LCM, consider the following resources:

Math Is Fun - Least Common Multiple Khan Academy - Least Common Multiple Purplemath - Finding the Least Common Multiple