Simplifying The Expression (3x + 4)(x - 7) A Step-by-Step Guide

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In mathematics, simplifying algebraic expressions is a fundamental skill. It involves rewriting an expression in a more concise and manageable form while maintaining its original value. This often involves applying the distributive property, combining like terms, and performing other algebraic manipulations. In this article, we will delve into simplifying the expression (3x + 4)(x - 7) step-by-step, providing a clear understanding of the process and the underlying principles. Mastering this skill is crucial for success in algebra and higher-level mathematics.

Understanding the Distributive Property

The distributive property is a cornerstone of simplifying expressions that involve parentheses. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In other words, you can multiply a term outside the parentheses by each term inside the parentheses and then add the results. The distributive property is not limited to the form a(b + c); it can also be applied when there are more terms inside the parentheses or when the expression involves subtraction, such as a(b - c) = ab - ac. The key is to ensure that each term within the parentheses is multiplied by the term outside. This principle extends to expressions where two binomials are multiplied together, like the one we are tackling today.

Step-by-Step Simplification of (3x + 4)(x - 7)

Let's break down the simplification of the expression (3x + 4)(x - 7) into manageable steps. This process, often referred to as the FOIL method (First, Outer, Inner, Last), is a systematic way to apply the distributive property when multiplying two binomials. Here's how it works:

1. First: Multiply the First Terms

Start by multiplying the first terms of each binomial: 3x and x. This gives us 3x * x = 3x². The term 3x² becomes the first part of our simplified expression. Remember, when multiplying variables with exponents, you add the exponents. In this case, x has an implied exponent of 1, so x¹ * x¹ = x^(1+1) = x².

2. Outer: Multiply the Outer Terms

Next, multiply the outer terms of the expression, which are 3x and -7. This gives us 3x * (-7) = -21x. The term -21x represents the contribution from the outer multiplication and is a crucial part of our simplification process. Be mindful of the negative sign; it's essential for accurate calculations.

3. Inner: Multiply the Inner Terms

Now, multiply the inner terms, which are 4 and x. This results in 4 * x = 4x. The term 4x is another component that will be combined with other like terms later in the process. Keep track of each term as you calculate it to ensure you don't miss any parts of the expression.

4. Last: Multiply the Last Terms

Finally, multiply the last terms of each binomial: 4 and -7. This gives us 4 * (-7) = -28. The term -28 is the constant term in our expanded expression. Again, pay close attention to the negative sign.

5. Combine the Terms

After performing all the multiplications, we have the expression 3x² - 21x + 4x - 28. The next step is to combine like terms to simplify the expression further. Like terms are terms that have the same variable raised to the same power. In this case, -21x and 4x are like terms.

Combining -21x and 4x, we get -21x + 4x = -17x. This simplifies our expression to 3x² - 17x - 28. This is the simplified form of the original expression.

Identifying the Correct Answer

Now that we have simplified the expression to 3x² - 17x - 28, we can compare it to the given options:

A. 3x² - 25x - 28 B. 3x² - 17x - 11 C. 3x² - 17x - 28 D. 3x² - 25x - 11

By comparing our simplified expression to the options, we can see that the correct answer is C. 3x² - 17x - 28.

Common Mistakes to Avoid

When simplifying expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and improve your accuracy. Here are some key mistakes to watch out for:

  • Sign Errors: One of the most frequent errors is making mistakes with signs, especially when dealing with negative numbers. For example, incorrectly multiplying or adding negative terms can lead to an incorrect result. Always double-check your signs and ensure you are applying the rules of arithmetic correctly.
  • Incorrect Distribution: Another common mistake is not distributing properly. Remember that each term inside the parentheses must be multiplied by the term outside. Forgetting to multiply one of the terms can lead to an incorrect simplification. Use the FOIL method systematically to ensure you cover all terms.
  • Combining Unlike Terms: Only like terms can be combined. Like terms have the same variable raised to the same power. For example, 3x² and -17x are not like terms and cannot be combined. Make sure you are only combining terms that have the same variable and exponent.
  • Arithmetic Errors: Simple arithmetic errors, such as adding or multiplying numbers incorrectly, can also lead to incorrect answers. Take your time and double-check your calculations to minimize these errors. Using a calculator can be helpful for more complex calculations.

Practice Problems

To reinforce your understanding of simplifying expressions, here are a few practice problems:

  1. Simplify: (2x + 3)(x - 4)
  2. Simplify: (5x - 1)(2x + 2)
  3. Simplify: (x + 6)(x - 6)
  4. Simplify: (4x - 3)(3x - 2)

Working through these problems will help you become more comfortable with the process and improve your accuracy. Remember to use the FOIL method and combine like terms carefully.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. By understanding the distributive property and following a systematic approach, such as the FOIL method, you can simplify complex expressions with confidence. In this article, we walked through the step-by-step simplification of the expression (3x + 4)(x - 7), highlighting common mistakes to avoid and providing practice problems to reinforce your understanding. With practice, you can master this skill and excel in your math studies. Remember, the key is to be methodical, pay attention to detail, and double-check your work to ensure accuracy. This skill not only helps in solving algebraic problems but also builds a strong foundation for more advanced mathematical concepts.

By consistently practicing and applying these techniques, you can significantly enhance your algebraic skills and tackle more complex problems with ease. Mathematics, at its core, is about understanding patterns and applying logical steps to reach solutions. Simplifying expressions is a perfect example of this, where each step builds upon the previous one to arrive at the final, simplified form. So, keep practicing, keep learning, and continue to explore the fascinating world of mathematics!