Simplify (x+7)(x+3): A Step-by-Step Guide

Hey guys! Today, let's dive into a fundamental concept in algebra: multiplying binomials. Specifically, we're going to tackle the expression $(x+7)(x+3)$. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, making sure you understand not just how to do it, but also why it works. So, grab your pencil and paper, and let's get started!

Understanding the FOIL Method

When we talk about multiplying binomials, one of the most popular methods is the FOIL method. FOIL is an acronym that stands for First, Outer, Inner, Last. It’s a handy mnemonic device that helps us remember to multiply each term in the first binomial by each term in the second binomial. Think of it as a systematic way to distribute terms and ensure nothing gets left out. Mastering the FOIL method is crucial for efficiently expanding expressions like $(x+7)(x+3)$. But before we jump into the mechanics, let's understand why this method works. It's essentially an application of the distributive property, which states that $a(b+c) = ab + ac$. We're just applying this property twice! First, we distribute the $x$ from the first binomial across the second binomial, and then we distribute the $7$. This careful distribution ensures that we account for every possible product between the terms in the two binomials. Without a systematic approach like FOIL, it's easy to miss a term or make a mistake, especially as the expressions become more complex. Understanding the underlying principle of distribution is key to confidently tackling more advanced algebraic manipulations later on. So, with the FOIL method in mind, we can approach our problem with a clear strategy. Each letter in FOIL represents a specific multiplication we need to perform, and by following this sequence, we can systematically expand the expression. This approach is not just about getting the right answer; it's about building a solid foundation in algebraic thinking. Remember, math is not just about memorizing rules, but also about understanding why those rules work. So, as we go through the steps, focus not just on the mechanics but also on the logic behind each operation. This deeper understanding will empower you to tackle more challenging problems with confidence and ease. Now, let's move on to applying the FOIL method to our specific expression, $(x+7)(x+3)$, and see how it works in practice.

Applying the FOIL Method to (x+7)(x+3)

Okay, let’s put the FOIL method into action with our expression $(x+7)(x+3)$. Remember, FOIL stands for First, Outer, Inner, Last. We'll go through each step meticulously to ensure we don't miss anything. First, we multiply the First terms in each binomial: $x$ from the first binomial and $x$ from the second binomial. This gives us $x * x = x^2$. This is the first term of our expanded expression. Next, we multiply the Outer terms: $x$ from the first binomial and $3$ from the second binomial. This results in $x * 3 = 3x$. Keep this term in mind as we move to the next step. Now, let's multiply the Inner terms: $7$ from the first binomial and $x$ from the second binomial. This gives us $7 * x = 7x$. We're almost there! Finally, we multiply the Last terms in each binomial: $7$ from the first binomial and $3$ from the second binomial. This yields $7 * 3 = 21$. So, after applying the FOIL method, we have four terms: $x^2$, $3x$, $7x$, and $21$. But we're not done yet! The next crucial step is to combine like terms. Look for terms that have the same variable raised to the same power. In our case, we have $3x$ and $7x$, both of which are terms with $x$ raised to the power of $1$. Combining these like terms is like adding apples to apples; we're simply adding the coefficients (the numbers in front of the $x$). So, $3x + 7x = 10x$. Now we can rewrite our expression with the combined like terms: $x^2 + 10x + 21$. And that, my friends, is our simplified expression! We've successfully multiplied the binomials $(x+7)(x+3)$ using the FOIL method and combined like terms to arrive at the final answer. This process might seem like a lot of steps at first, but with practice, it will become second nature. The key is to remember the FOIL acronym and systematically apply each step. And don't forget the crucial step of combining like terms at the end! This step is essential for simplifying the expression and arriving at the most concise answer. Now, let's take a closer look at why combining like terms is so important and how it helps us simplify algebraic expressions.

Simplifying the Expression by Combining Like Terms

Okay, so we've used the FOIL method to expand $(x+7)(x+3)$ and got $x^2 + 3x + 7x + 21$. But as we discussed earlier, the job isn't quite done until we combine like terms. Why is this so important? Well, combining like terms is like tidying up your room – it makes everything neater and easier to understand. In mathematical terms, it simplifies the expression, making it easier to work with in future calculations. Think of each term as representing a different type of object. In our expression, $x^2$ represents a square with sides of length $x$, $x$ represents a line of length $x$, and the constant $21$ represents 21 individual units. We can't directly add squares and lines – they're different entities. But we can add lines together. That's what combining like terms is all about. In our expression, $3x$ and $7x$ are like terms because they both contain the variable $x$ raised to the same power (which is $1$ in this case). This means they represent the same type of object – lines of length $x$. We can combine them by simply adding their coefficients: $3 + 7 = 10$. So, $3x + 7x$ becomes $10x$. This simplifies our expression from four terms to three: $x^2 + 10x + 21$. This simplified form is much cleaner and easier to work with. Imagine trying to solve an equation with the unsimplified expression – it would be much more cumbersome. By combining like terms, we've reduced the complexity and made the expression more manageable. This skill is crucial not just for multiplying binomials but for all sorts of algebraic manipulations. Whether you're solving equations, factoring polynomials, or working with more complex functions, combining like terms is a fundamental step in simplifying expressions. It's like the basic grammar of algebra – you need to master it to communicate effectively in the language of math. So, always remember to look for like terms after you've expanded an expression. It's the final touch that transforms a messy collection of terms into a streamlined and elegant mathematical statement. Now that we've thoroughly explored the process of multiplying binomials and combining like terms, let's summarize our steps and state the final answer.

The Final Answer and Key Takeaways

Alright, guys, let's recap! We started with the expression $(x+7)(x+3)$ and walked through the process of multiplying these binomials. We used the FOIL method – First, Outer, Inner, Last – to systematically multiply each term in the first binomial by each term in the second binomial. This gave us $x^2 + 3x + 7x + 21$. Then, we combined like terms, specifically the $3x$ and $7x$ terms, to simplify the expression. Adding these together, we got $10x$. This crucial step of combining like terms brought us to our final, simplified answer: $x^2 + 10x + 21$. So, the answer to the question (x+7)(x+3) = oxed{x^2 + 10x + 21}. This whole process highlights a few key takeaways. First, the FOIL method is your friend when multiplying binomials. It provides a structured approach to ensure you don't miss any terms. Second, combining like terms is essential for simplifying expressions and arriving at the most concise answer. It's like the finishing touch that makes everything neat and tidy. Third, understanding the why behind the methods is just as important as knowing the how. Knowing that the FOIL method is based on the distributive property helps you understand why it works and makes it easier to remember. Finally, practice makes perfect! The more you practice multiplying binomials and combining like terms, the more comfortable and confident you'll become. So, don't be afraid to tackle more problems and hone your skills. This is a foundational concept in algebra, and mastering it will set you up for success in more advanced topics. You've got this! Remember the steps, understand the logic, and keep practicing. Soon, multiplying binomials will be a breeze! And now you have a solid understanding of how to approach similar problems in the future. Keep up the great work, and happy calculating!