Simplify (x+7)(x-4): A Step-by-Step Guide

by Alex Johnson 42 views

Let's break down how to simplify the expression (x+7)(x-4). This is a common type of problem in algebra, and mastering it will help you tackle more complex equations later on. We'll walk through each step, explaining the logic behind it so you can understand not just the how, but also the why. Understanding the 'why' makes problem-solving so much easier in the long run!

Understanding the Distributive Property

The key to simplifying expressions like (x+7)(x-4) is the distributive property, often remembered by the acronym FOIL: First, Outer, Inner, Last. This method ensures that each term in the first set of parentheses is multiplied by each term in the second set. Let's dive into each part of the FOIL method and apply it to our problem.

  • First: Multiply the first terms in each set of parentheses. In our case, it's x * x, which equals x². This is the very first step in expanding our expression, and it lays the groundwork for combining like terms later on. Always start by identifying the first terms correctly, as this sets the stage for the rest of the calculation. Getting this initial step right is crucial for arriving at the correct final answer.
  • Outer: Multiply the outer terms in the expression. That's x * -4, resulting in -4x. The outer terms are the ones farthest away from each other when you write out the expression. Paying attention to the signs (positive or negative) is particularly important here. A simple sign error can throw off the entire calculation, so double-check your work.
  • Inner: Multiply the inner terms. Here, it's 7 * x, which gives us 7x. The inner terms are those closest to each other in the middle of the expression. Make sure you're clear on which terms are the 'inner' ones to avoid mixing them up with the 'outer' terms. Correctly identifying and multiplying these terms is essential for the next step, where we'll combine like terms.
  • Last: Multiply the last terms in each set of parentheses. This is 7 * -4, which equals -28. This final multiplication completes the expansion process. Ensure you get the sign correct! A positive times a negative is always a negative. This completes the FOIL process, and we're ready to simplify further.

Applying FOIL to (x+7)(x-4)

Let's put it all together. Applying the FOIL method to (x+7)(x-4) gives us:

x² (First: x * x) - 4x (Outer: x * -4) + 7x (Inner: 7 * x) - 28 (Last: 7 * -4)

So, our expression now looks like: x² - 4x + 7x - 28. We're not done yet! The next step is to combine like terms to simplify further.

Combining Like Terms

Now that we've expanded the expression, we need to combine the 'like terms'. Like terms are those that have the same variable raised to the same power. In our expression, x² - 4x + 7x - 28, the like terms are -4x and +7x. Combining these terms means adding their coefficients (the numbers in front of the x). So, -4x + 7x becomes 3x.

Simplifying the Expression

After combining like terms, our expression simplifies to: x² + 3x - 28. This is the final simplified form of the original expression. There are no more like terms to combine, and we've successfully expanded and simplified the given expression.

Why is Simplifying Important?

Simplifying expressions is a fundamental skill in algebra and is used extensively in higher-level mathematics. Simplified expressions are easier to work with when solving equations, graphing functions, and performing other mathematical operations. Think of it like this: a simplified expression is like a well-organized toolbox. You can quickly find the tools you need without having to rummage through a jumbled mess. Mastering simplification techniques makes problem-solving more efficient and less prone to errors.

Common Mistakes to Avoid

  • Sign Errors: One of the most common mistakes is making errors with signs (positive and negative). Always double-check your signs when multiplying and combining terms.
  • Incorrectly Applying FOIL: Make sure you multiply each term in the first set of parentheses by each term in the second set. A missed multiplication can lead to an incorrect answer.
  • Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine x² and x.
  • Forgetting to Distribute: Ensure you distribute the negative sign correctly when dealing with subtraction. For example, -(x - 2) becomes -x + 2.

Practice Problems

To solidify your understanding, try simplifying these expressions:

  1. (x + 3)(x - 2)
  2. (2x - 1)(x + 4)
  3. (x - 5)(x - 5)

Working through practice problems is the best way to improve your skills and build confidence. The more you practice, the easier it will become to recognize patterns and avoid common mistakes.

Conclusion

Simplifying expressions like (x+7)(x-4) is a core skill in algebra. By understanding and applying the distributive property (FOIL) and combining like terms, you can confidently tackle these types of problems. Remember to pay attention to signs, avoid common mistakes, and practice regularly. With a solid understanding of these concepts, you'll be well-prepared for more advanced mathematical challenges. The correct answer to simplifying (x+7)(x-4) is D. x² + 3x - 28.

For further learning and practice, consider exploring resources like Khan Academy's algebra section on polynomial manipulation: Khan Academy Algebra. This external resource can provide additional examples and exercises to reinforce your understanding.