Expand & Simplify: (5x + 7)(3 - 2x) Step-by-Step

Dec 9, 2025 by Admin 49 views

Hey there, math enthusiasts and curious minds! Ever looked at an expression like (5x + 7)(3 - 2x) and wondered, "How on Earth do I even begin to unravel that?" Well, you're in the perfect spot because today, we're going to break down the process of expanding and simplifying algebraic expressions in a way that's super easy to understand. This isn't just about getting the right answer; it's about understanding the why behind each step, building a solid foundation for all your future mathematical adventures. Whether you're a student tackling algebra for the first time or just need a quick refresher, this guide is designed to make polynomial expansion feel less like a chore and more like a fun puzzle. We'll dive deep into the distributive property, get friendly with the FOIL method, and ultimately arrive at a beautifully simplified algebraic expression. So, grab a pen and paper, and let's conquer this together!

Why Even Bother with Polynomial Expansion, Guys?

Alright, let's get real for a sec. You might be asking yourself, "Why do I even need to know how to expand and simplify something like (5x + 7)(3 - 2x)? What's the point beyond a textbook problem?" That's a totally fair question, and I'm here to tell you that polynomial expansion isn't just some abstract concept confined to your algebra class. It's a fundamental skill that underpins so much of higher mathematics and even has applications in various real-world scenarios, believe it or not! Think about it: when you're dealing with areas, volumes, or even complex financial models, often you start with expressions that represent relationships between different variables. Sometimes these relationships are given in a factored form, like our example, and to truly understand their behavior, to graph them, or to solve for specific values, you absolutely need to expand them into their standard polynomial form. This process of expanding and simplifying algebraic expressions allows us to see the bigger picture, revealing the individual terms and their combined effects. It helps us identify patterns, simplify calculations, and transform complex-looking problems into more manageable ones. Without the ability to expand polynomials, you'd be stuck trying to navigate equations that are needlessly complicated, making it nearly impossible to solve for unknowns or analyze trends. This skill is the gateway to understanding quadratic equations, factoring, calculus, physics formulas, and even some aspects of computer science algorithms. So, while it might seem like a simple exercise in multiplication right now, mastering the distributive property and the FOIL method is akin to learning the alphabet before you can write a novel. It's foundational, incredibly versatile, and genuinely empowering for anyone looking to deepen their mathematical understanding. It teaches you precision, careful handling of positive and negative signs, and methodical problem-solving – skills that extend far beyond the math classroom. So, yes, we bother with it because it's a cornerstone of algebraic fluency and a crucial tool in your mathematical toolkit!

Getting Started: What's a Polynomial Anyway?

Before we jump into the nitty-gritty of expanding our specific expression, let's make sure we're all on the same page about what we're actually working with. We're talking about polynomials here, folks! Now, that might sound like a fancy, intimidating word, but honestly, it just describes a type of algebraic expression. At its core, a polynomial is an expression consisting of variables (like our 'x') and coefficients (the numbers multiplying the variables), combined using only addition, subtraction, multiplication, and non-negative integer exponents. Simple, right? For example, 5x, 7, 3, and -2x are all parts of polynomials. When we talk about specific types, we often use prefixes: a monomial has one term (like 5x or 7), a binomial has two terms (like 5x + 7 or 3 - 2x), and a trinomial has three terms (like x² + 2x + 1). In our problem, we're multiplying two binomials together. Each individual piece of an expression, separated by addition or subtraction, is called a term. So, in (5x + 7), 5x is one term and 7 is another. In (3 - 2x), 3 is a term and -2x is another term. The number in front of a variable, like the '5' in 5x or the '-2' in -2x, is called the coefficient. It tells you how many of that variable you have. And the number all by itself, like '7' or '3', is a constant term because its value doesn't change, unlike the variable 'x'. Understanding these basic building blocks is super important because when we expand and simplify, we'll be dealing with these terms individually, multiplying them, and then combining the ones that are alike. Knowing the lingo makes the whole process much clearer and helps you feel more confident as you work through the steps. So, keep these definitions in mind as we move forward – they're the foundation upon which we'll build our understanding of polynomial expansion!

The Core Concept: The Distributive Property (aka FOIL for Binomials!)

Alright, guys, this is where the magic really happens! The absolute heart and soul of expanding algebraic expressions, especially when you're multiplying two binomials, is the distributive property. You've probably seen it before, perhaps in a simpler form like a(b + c) = ab + ac. It essentially says that when you multiply a term by an expression in parentheses, you have to multiply that term by each and every term inside the parentheses. Think of it like sharing: the outside term has to be 'distributed' to everyone inside. Now, when we're multiplying two binomials, like our (5x + 7)(3 - 2x), we apply the distributive property twice. It means each term in the first binomial has to be multiplied by each term in the second binomial. To help us remember this systematic process for binomials, we use a super handy mnemonic called FOIL!

Let's break down what FOIL stands for:

First: Multiply the first terms of each binomial together.
Outer: Multiply the outermost terms of the two binomials.
Inner: Multiply the innermost terms of the two binomials.
Last: Multiply the last terms of each binomial together.

This method ensures you don't miss any of the four necessary multiplications. Each of these products will give you a new term, and once you have all four, you'll be ready for the next step: combining like terms. The FOIL method is incredibly powerful because it provides a clear, step-by-step guide to expanding binomials accurately. It helps prevent common errors by making sure every possible combination of multiplication between the two sets of parentheses is accounted for. Mastering FOIL isn't just about memorizing an acronym; it's about internalizing the distributive property in a structured way for this specific type of problem. So, when you see two binomials side-by-side, your brain should immediately think: "FOIL time!" This is the critical step that transforms a compact, factored expression into a longer, expanded form that's easier to work with for further simplification. Pay close attention to this, because a solid understanding here will make the rest of the problem a breeze!

Let's Tackle Our Problem: (5x + 7)(3 - 2x) - Step-by-Step Breakdown!

Alright, folks, it's showtime! We've covered the why and the what, and now it's time to put our knowledge into action with our specific expression: (5x + 7)(3 - 2x). This is where we apply the FOIL method we just discussed, meticulously going through each step to ensure we expand it correctly. Remember, the goal here is to be thorough and precise, paying close attention to every detail, especially those pesky positive and negative signs. By breaking it down into smaller, manageable parts, we'll see that what looks like a complex algebraic expression is actually quite straightforward to handle. Each step builds on the last, so let's walk through it together, making sure we understand the logic behind every single multiplication and combination. This methodical approach is key to consistently getting the right answer and building confidence in your algebraic manipulation skills. So, let's grab our metaphorical magnifying glass and dive into the expansion of this binomial product!

Step 1: Apply the FOIL Method

Okay, let's roll up our sleeves and systematically apply the FOIL method to (5x + 7)(3 - 2x). This is the most crucial part of expanding algebraic expressions, as any mistake here will ripple through the entire problem. We're going to identify the 'First,' 'Outer,' 'Inner,' and 'Last' terms and multiply them one by one. Take your time with each multiplication, paying extra attention to the signs, because a common error here can derail the whole process. Remember, we treat each number and variable combination, including its sign, as a single term. So, (5x + 7) means positive 5x and positive 7. (3 - 2x) means positive 3 and negative 2x. Let's start breaking it down:

First terms: We multiply the first term of the first binomial by the first term of the second binomial. In our case, that's 5x multiplied by 3. This gives us: 5x * 3 = 15x. Simple enough, right? This establishes our first expanded term.
Outer terms: Next, we multiply the outermost term of the first binomial by the outermost term of the second binomial. This means we're taking 5x and multiplying it by -2x. Be careful with the negative sign here! 5x * (-2x) = -10x². See how the x's multiply to give x²? This is a really important detail that often trips people up. Don't forget those exponents!
Inner terms: Now, let's move to the innermost terms. We multiply the second term of the first binomial by the first term of the second binomial. This is 7 multiplied by 3. Both are positive, so it's straightforward: 7 * 3 = 21. This term is a constant, as it doesn't have an 'x' attached to it.
Last terms: Finally, we multiply the last term of the first binomial by the last term of the second binomial. This is 7 multiplied by -2x. Again, watch that negative sign! 7 * (-2x) = -14x. Just like our 'Outer' term, this one also has an 'x' component, which means it might be combined with other 'x' terms later on.

So, after applying the FOIL method, we have successfully expanded our product into four distinct terms. We haven't combined anything yet, just performed the initial multiplications. Each step here is crucial for laying the groundwork for the final simplified expression. Make sure you're comfortable with how each of these four products was derived before moving on. This careful execution of the distributive property via FOIL is the cornerstone of the entire process.

Step 2: Write Out the Expanded Expression

Alright, now that we've meticulously applied the FOIL method in Step 1, we have four individual terms from those multiplications. The next critical step in our journey to simplifying algebraic expressions is to neatly write out the entire expanded expression, combining these four results with their correct signs. This might seem like a small, obvious step, but trust me, it's where careful organization pays off big time! Listing everything out clearly helps us visualize all the parts we're working with and sets us up perfectly for the final simplification. From our previous step, we got these terms:

From First: 15x
From Outer: -10x²
From Inner: 21
From Last: -14x

Now, we just string them all together, maintaining their respective positive or negative signs. So, our expanded expression looks like this:

15x - 10x² + 21 - 14x

See how that looks? Each term is included exactly as we calculated it, with its sign correctly placed. It's super important to not drop any signs or terms here. If a term was negative, it remains negative in the expanded form. If it was positive, it remains positive. This intermediate step is like compiling all the ingredients before you start cooking. You need to make sure you have everything accounted for before you mix and combine. This expanded form, though not yet simplified, is the direct result of our polynomial expansion. It shows all the individual components that arose from multiplying our two binomials. Getting this step right ensures that you have all the correct pieces to move forward. Take a moment to double-check your work: did you write down all four terms? Are the signs correct? If everything looks good, then you're ready for the grand finale – combining like terms to achieve the most simplified form!

Step 3: Simplify by Combining Like Terms

Fantastic, guys! We're in the home stretch now. We've successfully expanded our binomials into 15x - 10x² + 21 - 14x. The final, crucial step in simplifying algebraic expressions is to combine what we call "like terms." This means grouping together any terms that have the exact same variable part and exponent. Think of it like sorting laundry: you wouldn't mix your socks with your shirts, right? Similarly, you can only add or subtract terms that are fundamentally the same. For example, 'x' terms can only be combined with other 'x' terms, 'x²' terms with other 'x²' terms, and constant numbers with other constant numbers. You absolutely cannot combine an 'x' term with an 'x²' term, or an 'x' term with a constant, because they are fundamentally different! They're like apples and oranges.

Let's identify the like terms in our expanded expression: 15x - 10x² + 21 - 14x.

Terms with x²: We only have one term with x²: -10x². There's nothing else to combine it with, so it stands alone for now.
Terms with x: We have 15x and -14x. These are like terms because they both have 'x' raised to the power of 1. We can combine them: 15x - 14x = 1x, or simply x.
Constant terms: We only have one constant term: 21. Like the x² term, it stands alone.

Now, we'll write out the simplified expression by combining these terms. It's standard practice to write polynomials in descending order of their exponents, meaning the term with the highest exponent comes first, then the next highest, and so on, ending with the constant term. So, our -10x² term will come first, followed by our 'x' term, and finally our constant term.

Putting it all together, our fully simplified algebraic expression is:

-10x² + x + 21

And there you have it! From a factored product of two binomials, we've successfully expanded and simplified it into a neat, standard quadratic trinomial. This process of identifying and combining like terms is essential for presenting your final answer in the most concise and usable form. Double-check your arithmetic when combining coefficients, and always ensure you're only combining terms that truly are