Factoring The Difference Of Monomials A And B A Step By Step Guide

In this comprehensive guide, we will delve into the process of factoring the difference of two monomials, A and B, where A is equal to 125 and B is equal to 27p^12. This problem falls under the realm of algebraic factorization, a fundamental concept in mathematics. We will explore the underlying principles and techniques involved in arriving at the factored form of A - B, providing a step-by-step explanation to enhance your understanding.

Understanding the Problem

Before we embark on the factorization process, let's first understand the problem statement thoroughly. We are given two monomials, A and B, defined as follows:

• A = 125
• B = 27p^12

Our objective is to determine the factored form of the expression A - B. This involves expressing A - B as a product of simpler algebraic expressions, typically polynomials. The factored form provides valuable insights into the structure and properties of the original expression.

Key Concepts and Techniques

To factor the difference of monomials A - B, we will leverage the concept of the difference of cubes. The difference of cubes is a special algebraic identity that allows us to factor expressions of the form a^3 - b^3. The identity is stated as follows:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

This identity provides a direct way to factor expressions that can be expressed as the difference of two perfect cubes. In our case, we can rewrite A and B as cubes:

• A = 125 = 5^3
• B = 27p^12 = (3p⁴⁾3

By recognizing A and B as perfect cubes, we can apply the difference of cubes identity to factor A - B.

Step-by-Step Factorization

Now, let's proceed with the step-by-step factorization of A - B using the difference of cubes identity:

1. Identify a and b:

   • a = 5
   • b = 3p^4

2. Apply the difference of cubes identity:

   A - B = a^3 - b^3 = (a - b)(a^2 + ab + b^2)

3. Substitute the values of a and b:

   (5 - 3p⁴⁾⁽⁵2 + 5 * 3p^4 + (3p⁴⁾2)

4. Simplify the expression:

   (5 - 3p^4)(25 + 15p^4 + 9p^8)

Therefore, the factored form of A - B is (5 - 3p^4)(25 + 15p^4 + 9p^8).

Verification of the Factored Form

To ensure the correctness of our factorization, we can verify the factored form by expanding it and comparing it with the original expression A - B. Expanding the factored form, we get:

(5 - 3p^4)(25 + 15p^4 + 9p^8) = 5(25 + 15p^4 + 9p^8) - 3p^4(25 + 15p^4 + 9p^8)

= 125 + 75p^4 + 45p^8 - 75p^4 - 45p^8 - 27p^12

= 125 - 27p^12

This matches the original expression A - B, confirming the accuracy of our factorization.

Alternative Approaches

While the difference of cubes identity provides a direct approach to factoring A - B, other methods can also be employed. One such method involves polynomial long division. However, this method is generally more complex and time-consuming compared to the difference of cubes identity.

Another alternative approach involves recognizing the expression as a special case of a more general factorization pattern. For instance, A - B can be viewed as a difference of two terms raised to the power of 3. However, this approach often requires a deeper understanding of algebraic manipulation and pattern recognition.

Common Mistakes to Avoid

When factoring the difference of monomials, it is crucial to avoid common mistakes that can lead to incorrect results. Some of the common mistakes include:

• Incorrectly applying the difference of cubes identity: Ensure that the identity is applied correctly, paying attention to the signs and terms involved.
• Forgetting to simplify the factored form: After applying the identity, simplify the resulting expression to its simplest form.
• Making arithmetic errors: Double-check all calculations to avoid arithmetic errors that can affect the final result.
• Failing to verify the factored form: Always verify the factored form by expanding it and comparing it with the original expression.

By being mindful of these common mistakes, you can improve your accuracy and confidence in factoring the difference of monomials.

Practice Problems

To solidify your understanding of factoring the difference of monomials, let's consider a few practice problems:

1. Factor the expression 64x^3 - 1.
2. Factor the expression 8a^6 - 27b^3.
3. Factor the expression 125m^9 - 216n^6.

By working through these practice problems, you can reinforce your skills and gain further proficiency in factoring the difference of monomials.

Conclusion

In this comprehensive guide, we have explored the process of factoring the difference of monomials A and B, where A = 125 and B = 27p^12. We have demonstrated the application of the difference of cubes identity to factor A - B, providing a step-by-step explanation to enhance your understanding. Additionally, we have discussed alternative approaches, common mistakes to avoid, and practice problems to further solidify your knowledge.

Factoring the difference of monomials is a fundamental concept in algebra with numerous applications in various mathematical contexts. By mastering this skill, you will be well-equipped to tackle more complex algebraic problems and enhance your overall mathematical proficiency. Remember to practice regularly and apply the concepts learned in this guide to a variety of problems to further strengthen your understanding. With consistent effort and a solid grasp of the underlying principles, you can confidently factor the difference of monomials and excel in your algebraic endeavors.

Therefore, the correct answer is A. (5 - 3p^4)(25 + 15p^4 + 9p^8).