Simplifying Polynomials Multiplying (3ab - 4b)(7a + 6b)

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Introduction to Polynomial Multiplication

In this comprehensive guide, we delve into the realm of algebraic expressions, specifically focusing on the multiplication of binomials. Our central task is to perform the indicated operation and simplify the expression (3ab - 4b)(7a + 6b). This problem falls under the category of polynomial multiplication, a fundamental skill in algebra. Understanding how to multiply polynomials is crucial for various mathematical applications, including solving equations, graphing functions, and modeling real-world scenarios. This guide will provide a step-by-step breakdown of the multiplication process, ensuring clarity and comprehension for learners of all levels. Polynomial multiplication involves distributing each term of one polynomial across all terms of the other polynomial. In simpler terms, it's like applying the distributive property multiple times. This process is crucial in simplifying algebraic expressions and solving equations. Mastering polynomial multiplication is essential for students as it forms the foundation for more advanced algebraic concepts.

Before diving into the specifics, let's briefly touch upon some key terminology. A binomial is a polynomial with two terms. In our expression, both (3ab - 4b) and (7a + 6b) are binomials. The terms within these binomials involve variables (a and b) and coefficients (the numerical values multiplying the variables). The goal of simplifying the expression is to expand the product of these binomials and combine any like terms. Like terms are terms that have the same variables raised to the same powers. For instance, terms with 'ab' can be combined, as can terms with 'b' or 'a'. This process of combining like terms is what ultimately leads to the simplified form of the expression. Throughout this guide, we will emphasize the importance of careful distribution and accurate arithmetic to achieve the correct simplified form. Understanding the basic principles of polynomial multiplication not only aids in solving mathematical problems but also enhances problem-solving skills applicable in various fields. This introduction sets the stage for a detailed exploration of the process, ensuring a solid grasp of the underlying concepts.

Step-by-Step Solution Using the Distributive Property

The cornerstone of multiplying binomials lies in the distributive property. This property states that for any numbers a, b, and c: a(b + c) = ab + ac. We extend this principle to polynomials, ensuring each term in the first binomial is multiplied by each term in the second binomial. Let's break down the process for (3ab - 4b)(7a + 6b).

  1. Distribute the first term of the first binomial (3ab) over the second binomial (7a + 6b):

    • 3ab * (7a + 6b) = (3ab * 7a) + (3ab * 6b)
    • = 21a²b + 18ab²

    In this first step, we're taking the term 3ab and multiplying it individually by both 7a and 6b. This is a direct application of the distributive property. The resulting terms, 21a²b and 18ab², are the products of these multiplications. It's crucial to keep track of the exponents when multiplying variables; for example, a * a becomes a². This initial distribution sets the stage for the next part of the process.

  2. Distribute the second term of the first binomial (-4b) over the second binomial (7a + 6b):

    • -4b * (7a + 6b) = (-4b * 7a) + (-4b * 6b)
    • = -28ab - 24b²

    Here, we repeat the distribution process but with the second term of the first binomial, -4b. We multiply -4b by both 7a and 6b, being careful to maintain the negative sign. The resulting terms are -28ab and -24b². Note that the multiplication of -4b and 7a yields -28ab, and -4b multiplied by 6b gives -24b². Paying close attention to the signs is essential for accurate calculations. This step completes the distribution of all terms, setting us up for the final simplification.

  3. Combine the results from steps 1 and 2:

    • (21a²b + 18ab²) + (-28ab - 24b²)
    • = 21a²b + 18ab² - 28ab - 24b²

    This step is about bringing together the results from the previous two distributions. We simply add the expanded forms obtained in steps 1 and 2. At this point, the expression may seem complex, but the next step involves identifying and combining like terms, which will simplify the expression considerably. This collection of terms represents the fully expanded form of the original product.

  4. Identify and combine like terms:

    • There are no like terms for 21a²b and -24b².
    • There are no like terms for 18ab² and -28ab.
    • However, there are terms with 'ab' but are not the same, so can't be simplified.
    • Final Result: 21a²b + 18ab² - 28ab - 24b²

    The final step in simplifying the expression is to identify and combine any like terms. Like terms are terms that have the same variables raised to the same powers. In this case, we look for terms with the same variable combinations (e.g., a²b, ab², ab, b²). Upon inspection, we find that there are no like terms that can be combined further. The term 21a²b is unique, as are 18ab², -28ab, and -24b². Since there are no like terms to combine, the expression is now fully simplified. The final result, 21a²b + 18ab² - 28ab - 24b², represents the product of the original binomials in its simplest form. This completes the multiplication and simplification process. Understanding each step, from distribution to combining like terms, is crucial for mastering polynomial multiplication.

Alternative Method: The FOIL Method

Another popular technique for multiplying binomials is the FOIL method, an acronym that stands for First, Outer, Inner, Last. This method provides a structured way to ensure all terms are multiplied correctly. While it’s essentially a mnemonic for the distributive property applied to binomials, it can be helpful for visual learners. Let's apply the FOIL method to our expression (3ab - 4b)(7a + 6b).

  1. First: Multiply the first terms of each binomial:

    • (3ab)(7a) = 21a²b

    The first step in the FOIL method involves multiplying the first terms of each binomial. In our case, these are 3ab and 7a. Multiplying these terms together gives us 21a²b. Remember to multiply both the coefficients (3 and 7) and the variables (ab and a). The product of a and a is a², so the result is 21a²b. This step ensures that we account for the product of the leading terms in each binomial.

  2. Outer: Multiply the outer terms of the expression:

    • (3ab)(6b) = 18ab²

    Next, we multiply the outer terms, which are the terms on the extreme ends of the expression: 3ab from the first binomial and 6b from the second binomial. Their product is 18ab². Again, we multiply the coefficients (3 and 6) and the variables (ab and b). The product of b and b is b², so the result is 18ab². This step ensures we've included the product of the outer terms in our expansion.

  3. Inner: Multiply the inner terms of the expression:

    • (-4b)(7a) = -28ab

    The third step involves multiplying the inner terms: -4b from the first binomial and 7a from the second binomial. Multiplying these gives us -28ab. It's crucial to pay attention to the negative sign in -4b. The product of -4 and 7 is -28, and the variables b and a multiply to ab. This step accounts for the product of the inner terms in our binomial expansion.

  4. Last: Multiply the last terms of each binomial:

    • (-4b)(6b) = -24b²

    Finally, we multiply the last terms of each binomial: -4b and 6b. Their product is -24b². Once again, we must consider the negative sign. The product of -4 and 6 is -24, and b multiplied by b is b². This step ensures that we've included the product of the last terms in our expansion.

  5. Combine all results:

    • 21a²b + 18ab² - 28ab - 24b²

    Now that we've multiplied all the terms using the FOIL method, we combine the results from each step: 21a²b, 18ab², -28ab, and -24b². Adding these terms together gives us the expanded form of the expression. The expression is now 21a²b + 18ab² - 28ab - 24b². The final step is to check for and combine any like terms to simplify the expression further. In this case, there are no like terms to combine.

  6. Simplify by combining like terms:

    • No like terms to combine.
    • Final Result: 21a²b + 18ab² - 28ab - 24b²

    After obtaining the expanded form, the final step is to simplify the expression by combining like terms. Like terms are those that have the same variables raised to the same powers. In our expression, we have terms with a²b, ab², ab, and b². A careful examination reveals that none of these terms are like terms, as they each have different variable combinations. Therefore, there are no terms to combine, and the expression is already in its simplest form. The final result is 21a²b + 18ab² - 28ab - 24b², which is the same result we obtained using the distributive property. The FOIL method, like the distributive property, is a systematic way to ensure all terms in the binomials are multiplied correctly, and this final simplification step confirms that we have reached the most reduced form of the expression. This method offers a clear, step-by-step approach to binomial multiplication.

Comparing the Distributive Property and FOIL Method

Both the distributive property and the FOIL method are effective ways to multiply binomials. The distributive property is the underlying principle, while FOIL is a mnemonic that organizes the distribution process specifically for binomials. The distributive property provides a foundational understanding of polynomial multiplication, highlighting how each term in one polynomial interacts with each term in another. It's a more general method that can be applied to polynomials of any size, not just binomials. This versatility makes it an essential concept in algebra. On the other hand, the FOIL method offers a structured approach that's easy to remember for binomial multiplication. FOIL's strength lies in its systematic breakdown of the multiplication process, ensuring that no term is missed. However, it's limited to binomials and doesn't directly extend to larger polynomials.

Choosing between these methods often comes down to personal preference and the specific problem at hand. The distributive property is more versatile and applicable in a broader range of scenarios, while FOIL provides a clear, step-by-step guide for binomials. Both methods lead to the same result when applied correctly. Whether one prefers the conceptual clarity of the distributive property or the mnemonic structure of FOIL, mastering both can enhance algebraic skills. Ultimately, understanding the relationship between these methods deepens the understanding of polynomial multiplication. Each approach reinforces the fundamental principles of algebra and helps build confidence in manipulating algebraic expressions. The choice of method becomes a matter of efficiency and comfort, with the goal of accurately simplifying expressions and solving mathematical problems. By mastering both techniques, learners are well-equipped to tackle a variety of algebraic challenges.

Common Mistakes to Avoid

When multiplying and simplifying algebraic expressions, certain mistakes are common. Identifying and understanding these potential pitfalls is crucial for accuracy. One frequent error is the incorrect application of the distributive property. This can involve either missing a term during distribution or multiplying terms out of order. Another common mistake is sign errors, especially when dealing with negative numbers. Failing to correctly account for negative signs during multiplication can lead to incorrect results. A third mistake is combining unlike terms. Only terms with the same variables raised to the same powers can be combined. Attempting to combine terms that don't meet this criterion will result in an incorrect simplification.

Additionally, students sometimes make mistakes when handling exponents. Remember that when multiplying terms with the same base, you add the exponents, but this rule doesn't apply when adding or subtracting terms. Another area of concern is the order of operations. It's important to follow the correct order (PEMDAS/BODMAS) to ensure accurate simplification. For example, multiplication should be performed before addition or subtraction. By being mindful of these common mistakes, one can significantly reduce errors in algebraic manipulations. Consistent practice and careful attention to detail are essential for mastering these skills. It is also helpful to double-check your work and use estimation techniques to verify that the final answer is reasonable. Understanding these pitfalls helps in building a strong foundation in algebra and enhances problem-solving abilities.

Practice Problems

To solidify your understanding, practice is essential. Here are a few problems similar to the example we worked through:

  1. (2x + 3)(x - 4)
  2. (5a - 2b)(3a + b)
  3. (4m + n)(2m - 3n)
  4. (p - 7)(p + 7)
  5. (3c - 5d)(3c + 5d)

Working through these problems will help you apply the distributive property and FOIL method. Remember to show your steps and check for like terms to combine. These practice problems cover various scenarios you might encounter, including binomials with different coefficients, variables, and signs. Consistent practice with these types of problems will help you build confidence and proficiency in algebraic manipulation. The more you practice, the more comfortable you will become with identifying patterns, applying the correct steps, and avoiding common mistakes. Each problem provides an opportunity to reinforce your understanding of the distributive property and the FOIL method, ensuring that you can accurately multiply and simplify binomial expressions. Practice also helps in developing problem-solving strategies and improving your overall mathematical skills.

Conclusion

In this guide, we have thoroughly explored the multiplication of binomials, focusing on the expression (3ab - 4b)(7a + 6b). We covered the distributive property, the FOIL method, common mistakes to avoid, and provided practice problems to reinforce your understanding. Mastering these techniques is crucial for success in algebra and beyond. By understanding and applying the distributive property and FOIL method, learners can effectively multiply binomials and simplify algebraic expressions. Avoiding common mistakes and engaging in regular practice are key to achieving proficiency in this fundamental algebraic skill.

This guide has provided a comprehensive approach to binomial multiplication, equipping learners with the necessary tools and knowledge to tackle similar problems with confidence. The ability to manipulate algebraic expressions is a cornerstone of mathematical education, and mastering these concepts opens doors to more advanced topics in algebra, calculus, and beyond. The consistent application of these techniques, coupled with a keen eye for detail, will ensure accuracy and efficiency in simplifying complex expressions. As learners continue to practice and refine their skills, they will develop a deeper appreciation for the elegance and power of algebraic manipulation. This understanding forms a solid foundation for future mathematical endeavors and enhances overall problem-solving abilities.