Find The Coefficient Of X⁴ In (2x+3)⁸ Using The Binomial Theorem

Hey guys! Let's dive into a classic math problem: finding the coefficient of a specific term in a binomial expansion. Specifically, we're going to use the Binomial Theorem to figure out the coefficient of x⁴ in the expansion of (2x + 3)⁸. Don't worry if this sounds intimidating at first; we'll break it down step by step, making it super easy to understand. This is a common problem in algebra and precalculus, and understanding it is key to mastering the Binomial Theorem. Ready to get started?

Unveiling the Binomial Theorem: The Foundation

So, what exactly is the Binomial Theorem? In a nutshell, it's a powerful formula that allows us to expand expressions of the form (a + b)ⁿ without actually having to multiply everything out. It's super handy when n is a large number, like in our case where n = 8. The theorem states:

(a + b)ⁿ = Σ (k=0 to n) [nCk * a⁽ⁿ⁻ᵏ⁾ * bᵏ]

Where:

• a and b are the terms within the binomial (in our case, 2x and 3).
• n is the power to which the binomial is raised (which is 8).
• k is the index of summation, ranging from 0 to n (so 0 to 8).
• nCk is the binomial coefficient, also written as ⁿCₖ or (n choose k), and is calculated as n! / (k! * (n - k)!). This represents the number of ways to choose k items from a set of n items. You can think of it as the coefficients that arise from the expansion. This notation is super important, so make sure you understand it!

In simpler terms, the theorem tells us how to find each term in the expansion. Each term is a product of three things: a binomial coefficient, a power of a, and a power of b. The summation sign (Σ) means we add up all these terms from k = 0 to n. It might seem like a lot to take in at once, but trust me, it's easier than it looks when you break it down, which is what we are going to do here!

Let’s translate this into something we can use. For our problem, a = 2x, b = 3, and n = 8. So, our expansion looks like:

(2x + 3)⁸ = Σ (k=0 to 8) [8Ck * (2x)⁽⁸⁻ᵏ⁾ * 3ᵏ]

Our mission is to find the term containing x⁴. That means we need to find the value of k that results in x⁴. In the term (2x)⁽⁸⁻ᵏ⁾, the power of x is (8 - k). So, we set 8 - k = 4 and solve for k. This means k = 4. Once we know k, we know exactly which term we need to calculate!

Pinpointing the Term with x⁴: The Calculation

Now that we know k = 4, we can focus on the specific term that contains x⁴. We plug k = 4 back into our formula:

Term with x⁴ = 8C₄ * (2x)⁽⁸⁻⁴⁾ * 3⁴

Let's break this down further and calculate each part. First, we need to find the binomial coefficient 8C₄. Using the formula n! / (k! * (n - k)!), we get:

8C₄ = 8! / (4! * (8 - 4)!) = 8! / (4! * 4!) = (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1) = 70

So, 8C₄ = 70. Now let's deal with the rest of the term:

(2x)⁽⁸⁻⁴⁾ = (2x)⁴ = 2⁴ * x⁴ = 16x⁴

And finally, 3⁴ = 81.

Putting it all together, the term with x⁴ is:

70 * 16x⁴ * 81

Now, we just need to multiply the numbers together to find the coefficient. So we have 70 * 16 * 81 = 90,720.

Therefore, the coefficient of x⁴ in the expansion of (2x + 3)⁸ is 90,720. That's a pretty big number, but hey, that's math for you!

Wrapping It Up: The Final Answer and Why It Matters

So, there you have it, guys! We've successfully used the Binomial Theorem to find the coefficient of x⁴ in the expansion of (2x + 3)⁸. The answer is 90,720. We broke down the problem into smaller, manageable steps, making it easier to understand and calculate. This approach is super useful for any Binomial Theorem problem.

Why does this matter? Well, the Binomial Theorem has tons of applications in mathematics, statistics, and even computer science. It's used in probability calculations, to model growth and decay, and in various algorithms. Understanding this theorem opens the door to more advanced concepts. Plus, it's a great way to boost your problem-solving skills and your understanding of algebra.

Remember, the key steps are:

1. Identify a, b, and n.
2. Determine the value of k that gives you the desired power of x.
3. Calculate the binomial coefficient (nCk).
4. Calculate the powers of a and b.
5. Multiply everything together.

That's it! You've successfully navigated a binomial expansion. Keep practicing, and you'll become a pro in no time! Remember to always double-check your calculations, especially when dealing with exponents and factorials. This will help you avoid silly mistakes. Practice makes perfect, and with each problem you solve, you'll gain more confidence and a deeper understanding of the Binomial Theorem. This is a core concept that builds a strong foundation for future mathematical studies. So, keep up the great work, and happy calculating!

I hope you found this guide helpful. If you have any questions, feel free to ask! Catch you later!