Expanding Polynomials: A Step-by-Step Guide

Hey guys! Let's dive into a common algebra problem: expanding the product of polynomials. Specifically, we're going to take the expression (3x + 1)(x + 5)(2x + 7) and rewrite it in the standard form ax³ + bx² + cx + d, where a, b, c, and d are just plain old integers. This process is super important because it helps us understand the structure of the expression and makes it easier to work with. So, buckle up, and let's get started. We'll break it down step-by-step so it's super easy to follow. Don't worry, even if polynomials feel a little intimidating at first, you'll see that with a bit of patience and some careful calculations, it's totally manageable. Ready? Let's go!

Step 1: Multiplying the First Two Binomials

Our first move in simplifying (3x + 1)(x + 5)(2x + 7) is to multiply the first two binomials: (3x + 1) and (x + 5). To do this, we'll use the distributive property, also known as the FOIL method. Remember that FOIL stands for First, Outer, Inner, Last – a neat trick to make sure you multiply every term correctly. Let’s do it:

• First: Multiply the first terms in each binomial: 3x * x = 3x²
• Outer: Multiply the outer terms: 3x * 5 = 15x
• Inner: Multiply the inner terms: 1 * x = x
• Last: Multiply the last terms: 1 * 5 = 5

Now, add all these terms together: 3x² + 15x + x + 5. We can simplify this a bit further by combining like terms. In this case, we have 15x and x, which can be combined to give us 16x. So, our simplified expression from multiplying the first two binomials is 3x² + 16x + 5. That wasn't so bad, right?

This first step is crucial because it reduces the initial problem of multiplying three binomials into a simpler problem involving a trinomial and a binomial. Getting this right sets you up for success in the rest of the problem. Always double-check your calculations in this step – a small mistake here will propagate through the rest of the process. Remember, precision is key! Keep an eye on the signs (+ and -) and make sure you're multiplying the coefficients correctly. If you're feeling a bit rusty, consider practicing with simpler examples first, like multiplying two binomials together. This will build your confidence and make the entire process feel more natural. And hey, don't forget the commutative and associative properties of multiplication – they're your friends!

Step 2: Multiplying the Result by the Third Binomial

Now that we've simplified the first two binomials, we have (3x² + 16x + 5) and we need to multiply this by the third binomial, which is (2x + 7). Again, we'll use the distributive property. This time, we're distributing each term of the trinomial across the binomial. It looks like a bit more work, but it’s still very systematic.

• Multiply 3x² by 2x and by 7: 3x² * 2x = 6x³ and 3x² * 7 = 21x²
• Multiply 16x by 2x and by 7: 16x * 2x = 32x² and 16x * 7 = 112x
• Multiply 5 by 2x and by 7: 5 * 2x = 10x and 5 * 7 = 35

Now, let's gather all these terms together: 6x³ + 21x² + 32x² + 112x + 10x + 35. The next step is to combine the like terms. We have:

• 21x² and 32x² which combine to 53x²
• 112x and 10x which combine to 122x

So, our final expanded form becomes 6x³ + 53x² + 122x + 35. Boom! We've done it! This step is all about making sure you multiply each term correctly and then patiently combining like terms. It's easy to lose track, so write everything out clearly and systematically. Double-check that you've accounted for every term. If you start to feel overwhelmed, break it down into smaller, more manageable steps. For example, first, focus on multiplying the first term of the trinomial by both terms of the binomial, then move to the second term, and so on. This approach will help you stay organized and make the process less daunting. It's also a great idea to regularly review the distributive property and the rules for exponents. These are the building blocks of this entire process, so a solid understanding of them is essential. With practice, you'll become more confident and accurate in your calculations. Trust me, it gets easier with each problem you solve!

Step 3: Identifying the Coefficients

We've successfully expanded the polynomial into the form 6x³ + 53x² + 122x + 35. Now we need to identify the coefficients a, b, c, and d in the standard form ax³ + bx² + cx + d. This is the easy part, guys! Just match the coefficients:

• a is the coefficient of x³, so a = 6
• b is the coefficient of x², so b = 53
• c is the coefficient of x, so c = 122
• d is the constant term, so d = 35

So there you have it! We've written (3x + 1)(x + 5)(2x + 7) in the form ax³ + bx² + cx + d, where a = 6, b = 53, c = 122, and d = 35. You've now mastered the art of expanding this type of polynomial expression. This skill is the cornerstone of many other algebraic concepts, so give yourself a pat on the back.

This final step is simple, but it's important for clarifying your result and demonstrating your understanding of the relationship between the expanded form and the coefficients. Always make sure to present your final answer clearly and concisely. Double-check your values to ensure they correspond to the correct terms in the expanded polynomial. Writing out a, b, c, and d separately also helps to reinforce your grasp of the concepts. And, don't be afraid to double-check your work using online tools or calculators! These can be helpful for verifying your solution and catching any minor errors. Most importantly, remember that practice makes perfect. The more problems you solve, the more comfortable and confident you'll become. Keep up the great work!

Conclusion: You Got This!

So there you have it, folks! We've successfully expanded (3x + 1)(x + 5)(2x + 7) into 6x³ + 53x² + 122x + 35. This whole process might seem a bit long, but trust me, with a little practice, it'll become second nature. Understanding how to expand polynomials is fundamental to your algebra journey. It's used in factoring, solving equations, and a whole bunch of other cool math stuff. Keep practicing, and don't hesitate to ask for help if you get stuck. You've got this!

This article provided a detailed, step-by-step guide to expanding polynomials. We hope this explanation helps. Remember, math is like any other skill – the more you practice, the better you get. So, keep at it, and don't be discouraged by mistakes. Instead, learn from them and keep moving forward. You'll be acing these problems in no time! Keep exploring and challenging yourself with more complex problems. You're building a strong foundation in algebra. Congrats on finishing this tutorial. If you need more help, you can use online calculators to check your answer. Keep practicing, and you'll get better and better.