Simplifying (2x - 9)(x + 6) A Step-by-Step Guide

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Simplifying algebraic expressions is a fundamental skill in mathematics, and mastering it can significantly enhance your problem-solving abilities. In this comprehensive guide, we will delve into the process of simplifying the expression (2x - 9)(x + 6). This type of problem often appears in algebra courses and standardized tests, making it crucial for students to understand the underlying principles and techniques. We will break down the steps involved, explain the reasoning behind each step, and provide additional insights to ensure a thorough understanding. Whether you're a student looking to improve your algebra skills or simply someone interested in mathematics, this guide will provide you with the knowledge and confidence to tackle similar problems.

Understanding the Distributive Property

To effectively simplify the expression (2x - 9)(x + 6), it is essential to grasp the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within parentheses. In simpler terms, it states that for any numbers a, b, and c:

  • a(b + c) = ab + ac

This property is the cornerstone of expanding and simplifying algebraic expressions, especially when dealing with binomials (expressions with two terms). In the given expression, we have two binomials, (2x - 9) and (x + 6), which need to be multiplied together. The distributive property provides the method for doing so systematically.

The core idea is that each term in the first binomial must be multiplied by each term in the second binomial. This ensures that we account for all possible combinations and arrive at the correct simplified form. This process might seem complex at first, but with practice, it becomes second nature. The distributive property is not just a mathematical trick; it’s a logical extension of how multiplication works. Understanding this logic will help you apply the property correctly in various contexts.

For instance, consider the simpler example of 3(x + 2). Using the distributive property, we multiply 3 by both x and 2, resulting in 3x + 6. This same principle applies to the more complex binomial multiplication we are addressing in this guide. Understanding this foundational principle is crucial for anyone looking to master algebraic simplification.

Applying the FOIL Method

When simplifying expressions like (2x - 9)(x + 6), a commonly used technique is the FOIL method. FOIL is an acronym that stands for First, Outer, Inner, Last, and it provides a structured way to ensure that each term in the first binomial is multiplied by each term in the second binomial.

Let's break down what each part of FOIL means:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the expression.
  • Inner: Multiply the inner terms in the expression.
  • Last: Multiply the last terms in each binomial.

By following this order, we systematically expand the expression and avoid missing any terms. The FOIL method is essentially a specific application of the distributive property, tailored for the multiplication of two binomials. It's a handy mnemonic device that makes the process more organized and less prone to errors.

Applying the FOIL method to our expression (2x - 9)(x + 6), we proceed as follows:

  1. First: Multiply the first terms: 2x * x = 2x²
  2. Outer: Multiply the outer terms: 2x * 6 = 12x
  3. Inner: Multiply the inner terms: -9 * x = -9x
  4. Last: Multiply the last terms: -9 * 6 = -54

This step-by-step approach ensures that we cover all the necessary multiplications. The FOIL method is not just about following a set of instructions; it’s about understanding the distributive property and applying it in a systematic way. This method is widely used because it simplifies the process and helps maintain accuracy. The result of this initial expansion is an unsimplified polynomial, which we will further simplify in the next step.

Step-by-Step Simplification of (2x - 9)(x + 6)

Now that we have applied the FOIL method to expand the expression (2x - 9)(x + 6), let's proceed with the step-by-step simplification process. Following the FOIL method, we obtained the following terms:

  • 2x² (First)
  • 12x (Outer)
  • -9x (Inner)
  • -54 (Last)

So, the expanded form of the expression is:

2x² + 12x - 9x - 54

The next crucial step in simplifying is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expanded expression, 12x and -9x are like terms because they both contain the variable 'x' raised to the power of 1. We can combine these terms by adding their coefficients.

Combining the like terms, we have:

12x - 9x = 3x

Now, we substitute this back into our expression:

2x² + 3x - 54

This is the simplified form of the original expression. We have combined all like terms, and there are no further simplifications possible. This final form is a quadratic expression, which is a polynomial of degree 2. The process of simplifying involves not just applying mechanical steps but also understanding the underlying algebraic principles. This step-by-step approach ensures clarity and accuracy, making it easier to arrive at the correct simplified expression.

Common Mistakes to Avoid

When simplifying algebraic expressions like (2x - 9)(x + 6), there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate simplification. Here are some of the most frequent errors:

  1. Incorrectly Applying the Distributive Property: A common mistake is failing to multiply each term in the first binomial by each term in the second binomial. This can lead to missing terms and an incorrect result. For instance, a student might forget to multiply the -9 by the x or the 6.
  2. Sign Errors: Errors with signs are very common, especially when dealing with negative numbers. For example, incorrectly multiplying -9 by 6 might result in +54 instead of -54. Paying close attention to the signs and double-checking the calculations can help prevent this.
  3. Combining Unlike Terms: Another frequent mistake is combining terms that are not like terms. For example, incorrectly adding 2x² and 3x. Remember, like terms must have the same variable raised to the same power to be combined.
  4. Forgetting to Distribute the Negative Sign: When dealing with expressions involving subtraction, it's crucial to distribute the negative sign correctly. For instance, if the expression were (2x - 9) - (x + 6), the negative sign must be distributed to both the x and the 6 in the second binomial.
  5. Rushing Through the Steps: Algebraic simplification requires careful attention to detail. Rushing through the steps can lead to careless errors. It’s better to take your time, write out each step clearly, and double-check your work.

By being mindful of these common mistakes and practicing diligently, you can improve your accuracy and confidence in simplifying algebraic expressions. Understanding the underlying principles and taking a methodical approach are key to success.

Practice Problems and Solutions

To solidify your understanding of simplifying expressions like (2x - 9)(x + 6), it’s essential to practice with additional problems. Here are a few practice problems along with their solutions:

Practice Problem 1: Simplify (3x + 2)(x - 4)

Solution:

  1. Apply the FOIL method:
    • First: 3x * x = 3x²
    • Outer: 3x * -4 = -12x
    • Inner: 2 * x = 2x
    • Last: 2 * -4 = -8
  2. Combine the terms: 3x² - 12x + 2x - 8
  3. Combine like terms: -12x + 2x = -10x
  4. Simplified expression: 3x² - 10x - 8

Practice Problem 2: Simplify (x - 5)(2x + 3)

Solution:

  1. Apply the FOIL method:
    • First: x * 2x = 2x²
    • Outer: x * 3 = 3x
    • Inner: -5 * 2x = -10x
    • Last: -5 * 3 = -15
  2. Combine the terms: 2x² + 3x - 10x - 15
  3. Combine like terms: 3x - 10x = -7x
  4. Simplified expression: 2x² - 7x - 15

Practice Problem 3: Simplify (4x - 1)(x + 7)

Solution:

  1. Apply the FOIL method:
    • First: 4x * x = 4x²
    • Outer: 4x * 7 = 28x
    • Inner: -1 * x = -x
    • Last: -1 * 7 = -7
  2. Combine the terms: 4x² + 28x - x - 7
  3. Combine like terms: 28x - x = 27x
  4. Simplified expression: 4x² + 27x - 7

By working through these practice problems, you reinforce your understanding of the FOIL method and the process of combining like terms. Each problem provides an opportunity to apply the steps and identify any areas where you may need further clarification. Consistent practice is key to mastering algebraic simplification.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying the expression (2x - 9)(x + 6) involves applying the distributive property and the FOIL method, combining like terms, and avoiding common mistakes. This comprehensive guide has walked you through each step, providing detailed explanations and examples to enhance your understanding. Mastering algebraic simplification is crucial for success in mathematics, as it forms the foundation for more advanced topics.

We began by understanding the distributive property, which is the fundamental principle behind expanding expressions. We then applied the FOIL method, a systematic way to multiply two binomials, ensuring that each term is correctly accounted for. The step-by-step simplification process highlighted the importance of combining like terms to arrive at the final answer: 2x² + 3x - 54. By breaking down the process into manageable steps, we demonstrated how to approach similar problems with confidence and accuracy.

Furthermore, we addressed common mistakes that students often make, such as sign errors and incorrectly combining terms. Awareness of these pitfalls is essential for avoiding errors and achieving correct solutions. The practice problems provided an opportunity to apply the learned techniques and reinforce the concepts.

By consistently practicing and applying these methods, you can develop a strong foundation in algebraic simplification. This skill will not only benefit you in your current studies but also in future mathematical endeavors. Remember, mathematics is a building process, and mastering the fundamentals is key to achieving higher levels of understanding and proficiency. With dedication and practice, you can confidently tackle more complex algebraic problems and excel in your mathematical journey.