Expand (3x+2)(5x-7): A Step-by-Step Guide

by Mei Lin 42 views

Hey everyone! Today, we're diving into a common algebraic problem: expanding and simplifying the expression (3x + 2)(5x - 7). This kind of problem is a staple in algebra, and mastering it will definitely help you in your math journey. We'll break it down step by step, making sure you understand each part of the process. So, grab your pencils and let's get started!

Understanding the Problem: Expanding Polynomials

Before we jump into the solution, let's quickly recap what it means to expand polynomials. When we see an expression like (3x + 2)(5x - 7), it means we need to multiply each term in the first set of parentheses by each term in the second set. Think of it as distributing each term across the other expression. There are a couple of methods we can use, but the most common one is the FOIL method. FOIL stands for First, Outer, Inner, Last, and it's a handy mnemonic to remember the order in which we multiply the terms. But don't worry if you prefer a different method; the key is to ensure every term gets multiplied correctly. We are essentially applying the distributive property multiple times. This property, a fundamental concept in algebra, states that a(b + c) = ab + ac. In our case, we're extending this principle to binomials (expressions with two terms). The goal here is to eliminate the parentheses and combine like terms to obtain a simplified polynomial expression. Understanding this foundational concept is crucial for tackling more complex algebraic problems later on.

Imagine you're building a house, and each term is a brick. To construct the house (the expanded expression), you need to carefully combine each brick in the correct order and manner. Similarly, in algebra, we meticulously multiply and combine terms to arrive at the simplified form. This process isn't just about following rules; it's about understanding how different parts of an expression interact with each other. By grasping the underlying logic, you'll be better equipped to handle various types of algebraic manipulations. Remember, practice makes perfect! The more you work with expanding polynomials, the more comfortable and confident you'll become.

Moreover, understanding polynomial expansion isn't just limited to academic exercises. It has practical applications in various fields, including engineering, physics, and computer science. For example, engineers might use polynomial expansion to model the behavior of circuits or mechanical systems. Physicists might use it to describe the motion of projectiles or the interactions of particles. Computer scientists might use it to develop algorithms for data analysis and machine learning. So, mastering this skill can open doors to exciting opportunities in the future.

Step-by-Step Solution Using the FOIL Method

Alright, let's break down the solution using the FOIL method. This method helps us systematically multiply the terms in the two binomials: (3x + 2)(5x - 7). Remember, FOIL stands for First, Outer, Inner, Last, guiding us through the multiplication process. Let's dive into each step:

  1. First: Multiply the first terms in each binomial. That's 3x from the first binomial and 5x from the second. So, 3x * 5x = 15x². This is where the exponents come into play. When multiplying variables with exponents, we add the exponents. In this case, x has an exponent of 1 (which is usually not written), so x¹ * x¹ = x¹⁺¹ = x².

  2. Outer: Multiply the outer terms. This means the first term of the first binomial (3x) and the last term of the second binomial (-7). So, 3x * -7 = -21x. Make sure you pay attention to the signs! A positive times a negative results in a negative.

  3. Inner: Multiply the inner terms. We're looking at the second term of the first binomial (2) and the first term of the second binomial (5x). So, 2 * 5x = 10x. This is a straightforward multiplication, but it's crucial not to overlook it.

  4. Last: Multiply the last terms in each binomial. That's 2 from the first binomial and -7 from the second. So, 2 * -7 = -14. Again, be mindful of the signs. A positive times a negative is a negative.

Now we have four terms: 15x², -21x, 10x, and -14. But we're not done yet! The next crucial step is to combine like terms.

Combining like terms is like sorting your socks after laundry. You group together the ones that are similar. In algebra, like terms are those that have the same variable raised to the same power. In our case, -21x and 10x are like terms because they both have x raised to the power of 1. We can combine them by simply adding their coefficients: -21 + 10 = -11. So, -21x + 10x = -11x.

Our expression now looks like this: 15x² - 11x - 14. We've successfully expanded and simplified the original expression! This final result is a quadratic trinomial, which is a polynomial with three terms and the highest power of the variable being 2.

Identifying the Correct Answer

Okay, we've gone through the expansion process step by step, and we've arrived at the simplified expression: 15x² - 11x - 14. Now, let's take a look at the answer choices provided and see which one matches our result.

The original question gave us these options:

  • (A) 15x² - x - 4
  • (B) 3x² + 15x - 11
  • (C) 15x² + 11x + 14
  • (D) 15x² - 11x - 14

By carefully comparing our result with the options, we can clearly see that (D) 15x² - 11x - 14 is the correct answer. It perfectly matches the expression we derived through the FOIL method and simplification.

It's important to note how the other options differ from the correct answer. Option (A) has the wrong coefficient for the x term and the constant term. Option (B) has incorrect coefficients for all terms. Option (C) has the correct coefficient for the x² term but the wrong signs for the x and constant terms. These differences highlight the importance of careful calculation and attention to detail when working with algebraic expressions. A small mistake in the multiplication or simplification process can lead to an incorrect answer.

When faced with multiple-choice questions like this, it's always a good idea to double-check your work. You can do this by either reviewing each step of your calculation or by plugging in a value for x into both the original expression and your simplified result. If the two expressions yield the same value, it's a good indication that your simplification is correct. This technique can be especially helpful in catching errors that you might have missed during the initial calculation.

Common Mistakes to Avoid

When expanding and simplifying expressions like (3x + 2)(5x - 7), there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you arrive at the correct answer. Let's take a look at some of these common errors:

  1. Forgetting to distribute: One of the most frequent mistakes is not multiplying every term in the first binomial by every term in the second binomial. Remember, each term in the first set of parentheses needs to be multiplied by each term in the second set. The FOIL method helps us remember this, but it's crucial to understand the underlying principle of distribution.

  2. Sign errors: Dealing with negative signs can be tricky. A common mistake is to incorrectly multiply or add negative numbers. For instance, 2 * -7 is -14, not 14. Similarly, -21x + 10x is -11x, not -31x. Pay close attention to the signs of each term and make sure you apply the rules of sign multiplication and addition correctly.

  3. Incorrectly combining like terms: Only terms with the same variable raised to the same power can be combined. For example, 15x² and -11x are not like terms because they have different powers of x. You can only combine terms like -21x and 10x. Make sure you identify the like terms correctly before attempting to combine them.

  4. Arithmetic errors: Simple arithmetic mistakes, such as miscalculating 3x * 5x as 8x² instead of 15x², can throw off the entire solution. It's a good idea to double-check your calculations, especially when dealing with larger numbers or multiple operations.

  5. Rushing through the steps: Algebra problems often require careful attention to detail. Rushing through the steps can lead to errors, such as skipping a term or making a mistake in the order of operations. Take your time, write down each step clearly, and double-check your work as you go along.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving algebraic problems. Remember, practice is key! The more you work with expanding and simplifying expressions, the better you'll become at avoiding these pitfalls.

Alternative Methods for Expansion

While the FOIL method is a popular and effective way to expand binomials, it's not the only method available. It's always good to have multiple tools in your toolbox, so let's explore some alternative approaches to expanding expressions like (3x + 2)(5x - 7).

  1. The Distributive Property (Box Method): This method is a visual approach that helps ensure every term is multiplied correctly. You create a grid (or a box) with the terms of one binomial across the top and the terms of the other binomial down the side. Then, you multiply the corresponding terms and fill in each cell of the grid. Finally, you combine like terms to get the simplified expression.

    For (3x + 2)(5x - 7), the grid would look like this:

    5x -7
    3x 15x² -21x
    2 10x -14

    Adding the terms within the grid gives us 15x² - 21x + 10x - 14, which simplifies to 15x² - 11x - 14.

  2. Vertical Multiplication: This method is similar to the way we multiply multi-digit numbers. You write one binomial below the other and multiply each term in the bottom binomial by each term in the top binomial, aligning like terms in columns. Then, you add the columns to get the final result.

        3x + 2
x   5x - 7
---------
  -21x - 14  (Multiplying by -7)
15x² + 10x   (Multiplying by 5x)
---------
15x² - 11x - 14 (Adding the columns)

  3. Direct Application of the Distributive Property: This method involves applying the distributive property twice. First, you distribute one of the binomials over the other. Then, you distribute again to eliminate the remaining parentheses.

    (3x + 2)(5x - 7) = 3x(5x - 7) + 2(5x - 7) = 15x² - 21x + 10x - 14 = 15x² - 11x - 14

Each of these methods has its advantages and disadvantages, and the best method for you might depend on your personal preference and the specific problem you're solving. The key is to choose a method that you understand well and can apply consistently and accurately.

Conclusion: Mastering Polynomial Expansion

Alright, guys, we've reached the end of our journey on evaluating the expression (3x + 2)(5x - 7). We've explored the problem, broken down the solution step-by-step using the FOIL method, identified the correct answer, discussed common mistakes to avoid, and even looked at alternative methods for expansion. Hopefully, you now have a solid understanding of how to tackle these types of problems.

Mastering polynomial expansion is a crucial skill in algebra, and it's one that will serve you well in many areas of mathematics and beyond. It's not just about memorizing the FOIL method or other techniques; it's about understanding the underlying principles of distribution and combining like terms. When you grasp these concepts, you'll be able to approach more complex algebraic problems with confidence.

Remember, practice makes perfect! The more you work with expanding and simplifying expressions, the more comfortable and proficient you'll become. Don't be afraid to make mistakes – they're a natural part of the learning process. Just learn from them, and keep practicing.

So, keep honing your skills, and you'll be expanding polynomials like a pro in no time! Good luck, and happy calculating!