Factoring 8x³ - 125 Find Values Of J And K

by Admin 43 views

The realm of algebra often presents us with intriguing challenges, and one such challenge lies in the factorization of expressions. In this comprehensive guide, we delve into the factorization of the expression 8x³ - 125, a classic example that showcases the application of algebraic identities. Our primary focus is to determine the values of 'j' and 'k' in the factorization (2x - 5)(jx² + kx + 25). This exploration will not only enhance your understanding of factorization techniques but also provide a step-by-step approach to solving similar algebraic problems.

The problem at hand involves factoring a difference of cubes, a special pattern in algebra. The expression 8x³ - 125 fits this pattern perfectly. Recognizing this pattern is the first crucial step in simplifying the expression. We are given that the factorization results in the form (2x - 5)(jx² + kx + 25), and our task is to find the values of the coefficients 'j' and 'k'. This task requires a solid understanding of algebraic identities and the ability to manipulate expressions effectively.

Recognizing the Difference of Cubes Pattern

The difference of cubes pattern is a fundamental algebraic identity that states:

a³ - b³ = (a - b)(a² + ab + b²)

This pattern is our key to unlocking the factorization of 8x³ - 125. To apply this pattern, we need to identify 'a' and 'b' in our expression. We can rewrite 8x³ as (2x)³ and 125 as 5³. This transformation allows us to clearly see that a = 2x and b = 5. Now, we can confidently apply the difference of cubes identity.

Applying the Difference of Cubes Identity

Substituting a = 2x and b = 5 into the difference of cubes identity, we get:

(2x)³ - 5³ = (2x - 5)((2x)² + (2x)(5) + 5²)

Now, let's simplify the expression on the right side:

(2x - 5)(4x² + 10x + 25)

This is the factored form of 8x³ - 125. Comparing this with the given factorization (2x - 5)(jx² + kx + 25), we can easily identify the values of 'j' and 'k'.

Now that we have the factored form of 8x³ - 125 as (2x - 5)(4x² + 10x + 25), we can directly compare it with the given form (2x - 5)(jx² + kx + 25). By comparing the coefficients of the corresponding terms, we can determine the values of 'j' and 'k'.

Comparing Coefficients

Comparing the two expressions:

(2x - 5)(4x² + 10x + 25) = (2x - 5)(jx² + kx + 25)

We can see that:

  • The coefficient of x² in the first expression is 4, and in the second expression, it is 'j'. Therefore, j = 4.
  • The coefficient of x in the first expression is 10, and in the second expression, it is 'k'. Therefore, k = 10.

Thus, we have successfully determined the values of 'j' and 'k' by applying the difference of cubes identity and comparing coefficients.

In conclusion, the factorization of 8x³ - 125 is indeed (2x - 5)(4x² + 10x + 25), which means the values of j and k are 4 and 10, respectively. This problem serves as a great example of how recognizing algebraic patterns, like the difference of cubes, can significantly simplify complex expressions. By applying the appropriate identities and carefully comparing coefficients, we can solve seemingly challenging problems with ease. This skill is invaluable in various mathematical contexts, making it an essential tool in any algebra enthusiast's arsenal.

Looking back at the options provided, we can now confidently select the correct answer. The values we found, j = 4 and k = 10, correspond to option D. Therefore, the correct answer is:

D. j = 4 and k = 10

This problem not only tests our ability to factorize expressions but also our attention to detail in selecting the correct answer from the given options. It reinforces the importance of understanding fundamental algebraic concepts and applying them systematically to solve problems.

To further enhance your understanding and problem-solving skills in algebra, here are some additional insights and tips related to factorization and algebraic identities:

Mastering Algebraic Identities

Algebraic identities are the backbone of factorization. Familiarize yourself with the most common identities, such as:

  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²
  • (a + b)(a - b) = a² - b²
  • (a + b)³ = a³ + 3a²b + 3ab² + b³
  • (a - b)³ = a³ - 3a²b - 3ab² - b³
  • a³ + b³ = (a + b)(a² - ab + b²)
  • a³ - b³ = (a - b)(a² + ab + b²)

Understanding these identities will enable you to quickly recognize patterns and simplify expressions efficiently.

Practice Regularly

Like any mathematical skill, factorization requires consistent practice. Solve a variety of problems to strengthen your understanding and develop your problem-solving intuition. Start with simpler problems and gradually move on to more complex ones. The more you practice, the more comfortable you will become with different factorization techniques.

Look for Common Factors

Before applying any identities, always look for common factors in the expression. Factoring out common factors can simplify the expression and make it easier to factorize further. For example, in the expression 2x² + 4x, you can factor out 2x to get 2x(x + 2).

Grouping Terms

In some cases, you may need to group terms to factorize an expression. This technique involves rearranging the terms and grouping them in such a way that you can factor out common factors from each group. For example, in the expression ax + ay + bx + by, you can group the terms as (ax + ay) + (bx + by) and then factor out common factors to get a(x + y) + b(x + y). Finally, you can factor out (x + y) to get (x + y)(a + b).

Verification

After factorizing an expression, always verify your answer by multiplying the factors back together. If you get the original expression, your factorization is correct. This step is crucial to ensure accuracy and avoid errors.

Seeking Help When Needed

If you encounter difficulties while factorizing expressions, don't hesitate to seek help from teachers, tutors, or online resources. There are many resources available to help you understand factorization techniques and solve problems effectively. Learning from others and collaborating can significantly enhance your understanding.

Factorization is not just an abstract mathematical concept; it has numerous real-world applications in various fields, including:

  • Engineering: Engineers use factorization to simplify complex equations and design structures, circuits, and systems.
  • Computer Science: Factorization is used in cryptography, data compression, and algorithm design.
  • Physics: Physicists use factorization to solve equations related to motion, energy, and forces.
  • Economics: Economists use factorization to model economic systems and analyze financial data.
  • Finance: Financial analysts use factorization to assess investment risks and make informed decisions.

Understanding factorization can open doors to various career paths and provide you with valuable problem-solving skills applicable in diverse contexts.

The factorization of 8x³ - 125 and the determination of j and k serve as a testament to the power and elegance of algebraic techniques. By mastering the difference of cubes identity and honing your problem-solving skills, you can tackle a wide range of algebraic challenges with confidence. Remember, practice is key to success in mathematics. Embrace the journey of learning, explore different problem-solving strategies, and never shy away from seeking help when needed. With dedication and perseverance, you can unlock the beauty and power of factorization and excel in your mathematical endeavors.

This exploration of factorization not only enhances your mathematical skills but also provides a foundation for understanding more advanced concepts in algebra and calculus. The ability to manipulate expressions, recognize patterns, and apply appropriate techniques is invaluable in various academic and professional pursuits. So, continue to explore the world of mathematics, embrace the challenges, and celebrate the triumphs along the way.