Multiplying Binomials A Step-by-Step Guide With (3rs - 7)(3rs - 5) Example
Hey guys! Today, we're diving into a common algebra problem: multiplying binomials. Specifically, we're going to tackle the expression (3rs - 7)(3rs - 5). Don't worry, it's not as intimidating as it looks! We'll break it down step by step, so you can easily follow along and master this skill. Multiplying binomials is a fundamental concept in algebra, and understanding it will help you in various mathematical contexts, including solving equations, simplifying expressions, and even in calculus later on. So, let's get started and make sure you're comfortable with this important algebraic technique!
Understanding Binomials
Before we jump into the multiplication process, let's quickly recap what binomials are. A binomial is simply an algebraic expression that consists of two terms. These terms are connected by either an addition or subtraction sign. For example, in our expression, 3rs - 7 and 3rs - 5 are both binomials. The first term in each binomial includes variables (r and s), while the second term is a constant. Recognizing binomials is the first step toward multiplying them correctly. Remember, these terms can involve variables, constants, or a combination of both, making binomials versatile components in algebraic expressions. Grasping the concept of binomials is crucial because they appear frequently in algebra and beyond. So, now that we've refreshed our understanding, let's move on to the method we'll use to multiply them.
The FOIL Method Explained
Now, let's talk about the method we'll use to multiply these binomials: FOIL. No, we're not wrapping leftovers – FOIL is an acronym that stands for First, Outer, Inner, Last. It's a handy way to remember the order in which we need to multiply the terms of the two binomials. This method ensures that every term in the first binomial is multiplied by every term in the second binomial. Think of it as a systematic way to distribute and expand the expression. Without a method like FOIL, it's easy to miss terms and get the wrong answer. So, let's break down what each part of FOIL means and how we'll apply it to our problem. Understanding FOIL is key to mastering binomial multiplication, making it an essential tool in your algebraic toolkit. Next, we'll apply the FOIL method to our specific binomials, (3rs - 7) and (3rs - 5), so you can see exactly how it works.
First Terms
The "F" in FOIL stands for First. This means we multiply the first terms of each binomial together. In our case, the first term of (3rs - 7) is 3rs, and the first term of (3rs - 5) is also 3rs. So, we multiply 3rs * 3rs. When multiplying these terms, we multiply the coefficients (the numbers) and add the exponents of the variables. Here, 3 multiplied by 3 gives us 9, and rs multiplied by rs gives us r^2s^2. Therefore, the product of the first terms is 9r^2s^2. This first step sets the foundation for the rest of the multiplication process. Getting the first terms right is crucial because it's often the term with the highest degree, so accuracy here is key to simplifying the final expression correctly. This part of the FOIL method is straightforward but important, so let’s make sure we’ve got it down before moving on.
Outer Terms
Next up is the "O" in FOIL, which stands for Outer. This means we multiply the outer terms of the binomials. Looking at (3rs - 7)(3rs - 5), the outer terms are 3rs from the first binomial and -5 from the second binomial. So, we need to multiply 3rs * -5. Multiplying these, we get -15rs. Remember to pay close attention to the signs! A negative multiplied by a positive results in a negative. The outer terms are often where students might make a mistake with signs, so it’s essential to be careful here. This step is just as important as the first, as it ensures we’re considering all the necessary combinations of terms. So, double-check your signs and coefficients, and let's move on to the next step.
Inner Terms
Now we tackle the "I" in FOIL, which represents the Inner terms. These are the terms inside the binomials that we haven't multiplied yet. In the expression (3rs - 7)(3rs - 5), the inner terms are -7 and 3rs. So, we multiply -7 * 3rs. This gives us -21rs. Again, keep a close eye on the signs. Multiplying a negative number by a positive term results in a negative term. The inner terms, like the outer terms, play a critical role in the final simplified expression. They often combine with the outer terms during the simplification process, so accuracy is paramount. Make sure you're comfortable identifying and multiplying the inner terms before moving on. Let's head to the last step in the FOIL method.
Last Terms
Finally, we have the "L" in FOIL, which stands for Last. This means we multiply the last terms of each binomial. In our example, the last term of (3rs - 7) is -7, and the last term of (3rs - 5) is -5. We multiply these together: -7 * -5. A negative times a negative gives us a positive, so the result is +35. This last step completes the distribution process, ensuring that every term in the first binomial has been multiplied by every term in the second binomial. The last terms are crucial for getting the constant term correct in the final simplified expression. With all parts of FOIL completed, we've successfully expanded the product of the binomials. Now, the next step is to combine like terms, which we'll cover in the following section.
Combining Like Terms
Okay, we've done the FOIL dance – now it's time to combine like terms. After applying the FOIL method to (3rs - 7)(3rs - 5), we have: 9r^2s^2 - 15rs - 21rs + 35. Like terms are those that have the same variables raised to the same powers. In our expression, -15rs and -21rs are like terms because they both have rs to the power of 1. To combine them, we simply add their coefficients. So, -15rs + (-21rs) equals -36rs. The other terms, 9r^2s^2 and +35, don't have any like terms, so they stay as they are. This step is crucial because it simplifies the expression to its most manageable form. Combining like terms not only makes the expression cleaner but also makes it easier to work with in further calculations or problem-solving. So, make sure you're comfortable identifying and combining like terms – it's a fundamental skill in algebra. Let's move on to writing our final simplified answer.
The Final Simplified Answer
Alright, after combining like terms, we've arrived at the final simplified answer. We started with (3rs - 7)(3rs - 5), applied the FOIL method, and combined like terms. The final expression is: 9r2s2 - 36rs + 35. This is the most simplified form of the product of our binomials. We've successfully multiplied the binomials and simplified the result by combining like terms. Remember, the order of terms might vary, but as long as the signs are correct, the expression is equivalent. The final answer represents the complete solution to our problem, showcasing the result of expanding and simplifying the original expression. So, give yourself a pat on the back – you've successfully navigated multiplying binomials! Now, let’s recap the steps we took to get here.
Recap of the Steps
Let's quickly recap the steps we took to solve this problem. First, we identified that we needed to multiply two binomials: (3rs - 7)(3rs - 5). Then, we used the FOIL method, which stands for First, Outer, Inner, Last, to multiply each term in the first binomial by each term in the second binomial. This gave us 9r^2s^2 - 15rs - 21rs + 35. Next, we combined like terms, specifically -15rs and -21rs, which resulted in -36rs. Finally, we wrote out the simplified expression: 9r^2s^2 - 36rs + 35. By following these steps, we ensured that we correctly multiplied the binomials and simplified the result. Remembering these steps will help you tackle similar problems with confidence. This process is a cornerstone of algebraic manipulation, and mastering it will benefit you in various mathematical contexts. So, keep practicing, and you'll become a pro at multiplying binomials!
Practice Problems
To really nail this skill, it's essential to practice. Try multiplying the following binomials on your own:
- (2x + 3)(x - 4)
- (5a - 2)(3a + 1)
- (4m + n)(m - 2n)
Work through these problems using the FOIL method and remember to combine like terms. Practice makes perfect, and the more you work with binomials, the more comfortable you'll become. These practice problems are designed to reinforce your understanding and give you hands-on experience with the process. Don't hesitate to review the steps we discussed earlier if you need a refresher. And most importantly, remember to double-check your work, especially the signs, to ensure you arrive at the correct simplified answer. So, grab a pencil and paper, and let's put your new skills to the test!
Conclusion
And there you have it! We've successfully multiplied the binomials (3rs - 7)(3rs - 5) and simplified the answer. Remember, the key is to use the FOIL method systematically and then combine like terms. Multiplying binomials is a fundamental skill in algebra, and mastering it will set you up for success in more advanced topics. Don't be afraid to practice and make mistakes – that's how you learn! Keep applying these steps, and you'll find that multiplying binomials becomes second nature. Whether you're solving equations, simplifying expressions, or tackling more complex algebraic challenges, this skill will be invaluable. So, keep up the great work, and remember, you've got this! If you found this guide helpful, feel free to share it with your friends and classmates. Happy multiplying, guys!
