Finding The Second Term In A Binomial Expansion Of (2r - 3s)^12

Hey guys! Ever find yourself staring blankly at binomial expansions, wondering how to tackle those pesky terms? Well, you're not alone! Let's break down a classic binomial expansion problem together. Today, we're diving into the expansion of (2r - 3s)^12 to find that elusive second term. Trust me, it's not as daunting as it seems. We'll take it step by step, so you'll be a binomial whiz in no time. We'll explore the binomial theorem, the formula behind it, and how to apply it practically. So, grab your thinking caps, and let's get started!

Understanding the Binomial Theorem

At the heart of solving this problem is the binomial theorem, a powerful tool for expanding expressions of the form (a + b)^n. Think of it as a shortcut that saves you from multiplying (a + b) by itself 'n' times. That could take ages, especially with higher powers like 12! The binomial theorem provides a systematic way to find each term in the expansion. The general formula looks like this:

(a + b)^n = Σ [nCk * a^(n-k) * b^k]

Where:

• 'n' is the power to which the binomial is raised.
• 'k' is the term number (starting from 0).
• 'Σ' means we're summing over all possible values of k (from 0 to n).
• 'nCk' represents the binomial coefficient, also known as "n choose k," which is calculated as n! / (k! * (n-k)!). The exclamation mark denotes the factorial, where a number is multiplied by all positive integers less than it (e.g., 5! = 5 * 4 * 3 * 2 * 1).

Let's break this down further. The binomial theorem essentially tells us that each term in the expansion is a product of three things: a binomial coefficient, 'a' raised to some power, and 'b' raised to some power. The binomial coefficient determines the numerical factor of the term, while the powers of 'a' and 'b' dictate the variables and their exponents. The magic lies in understanding how 'k' orchestrates this dance. As 'k' increases from 0 to 'n', we generate each successive term in the expansion. The exponent of 'b' is simply 'k', while the exponent of 'a' is 'n - k'. The binomial coefficient 'nCk' gives us the numerical coefficient for that particular term. Understanding this formula is the key to unlocking binomial expansions, so make sure you've got it down before we move on!

Identifying the Second Term

Now, let's get specific. In our problem, we need to find the second term in the expansion of (2r - 3s)^12. Remember that in binomial expansions, we start counting terms from 0, not 1. So:

• The first term corresponds to k = 0.
• The second term corresponds to k = 1.
• The third term corresponds to k = 2, and so on.

This is a crucial detail! Miscounting the term number is a common pitfall. Always remember that the term number is one more than the value of 'k'. So, for the second term, we're looking at k = 1. Now that we know 'k', we can plug it into the binomial theorem formula along with the other values from our problem. In this case, a = 2r, b = -3s, and n = 12. We've identified all the pieces of the puzzle. The next step is to carefully substitute these values into the formula and simplify. This is where attention to detail is paramount. A small mistake in the calculation can lead to a completely wrong answer. So, take your time, double-check your work, and you'll be on the right track. Let's move on to the actual calculation and see how it all comes together!

Applying the Formula to Find the Second Term

Okay, guys, time to put the binomial theorem into action! We know:

• a = 2r
• b = -3s
• n = 12
• k = 1 (for the second term)

Plugging these values into the formula, we get:

Term 2 = 12C1 * (2r)^(12-1) * (-3s)^1

Let's break this down piece by piece. First, we need to calculate the binomial coefficient 12C1, which is "12 choose 1." Using the formula nCk = n! / (k! * (n-k)!), we have:

12C1 = 12! / (1! * 11!) = 12

This means there are 12 ways to choose 1 item from a set of 12 items. Now, let's move on to the other parts of the term. We have (2r)^(12-1) which simplifies to (2r)^11. Remember that the exponent applies to both the 2 and the r, so we get 2^11 * r^11. Calculating 2^11 gives us 2048. Next, we have (-3s)^1, which is simply -3s. Now, we have all the pieces: 12, 2048 * r^11, and -3s. The final step is to multiply these together. This is where careful arithmetic is essential. We're dealing with relatively large numbers, so it's easy to make a mistake if we're not paying attention. Let's put it all together in the next section and see what the final answer is!

Calculating the Result

Alright, let's bring it all home! We've got the pieces, now we just need to multiply them together. From the previous step, we have:

Term 2 = 12 * (2048 * r^11) * (-3s)

First, let's multiply the numbers: 12 * 2048 * -3. This gives us -73,728. Then, we have the variables r^11 and s. So, putting it all together, the second term in the binomial expansion is:

-73,728 r^11 s

And there you have it! We've successfully navigated the binomial theorem and found the second term. The key was breaking down the problem into manageable steps: understanding the formula, identifying the correct values, plugging them in, and carefully calculating the result. It might seem like a lot at first, but with practice, these problems become much more straightforward. Now, let's take a moment to reflect on what we've learned and how we can apply this knowledge to other binomial expansion problems.

Conclusion: Mastering Binomial Expansions

So, guys, we've cracked the code on finding the second term in the binomial expansion of (2r - 3s)^12. We've journeyed through the binomial theorem, dissected its formula, and applied it step-by-step. Remember, the binomial theorem is your best friend when dealing with expansions of the form (a + b)^n. It saves you tons of time and effort compared to manual multiplication. The key takeaways from this exercise are:

1. Understanding the Binomial Theorem Formula: Make sure you're comfortable with the formula and what each symbol represents.
2. Identifying the Term Number: Remember to start counting terms from k = 0. The second term corresponds to k = 1, and so on.
3. Careful Calculation: Pay close attention to detail when substituting values and performing the arithmetic. A small mistake can lead to a wrong answer.
4. Practice Makes Perfect: The more binomial expansion problems you solve, the more comfortable you'll become with the process.

By mastering these concepts, you'll be well-equipped to tackle a wide range of binomial expansion problems. So, don't shy away from them! Embrace the challenge, and remember to break down complex problems into smaller, more manageable steps. Keep practicing, and you'll be a binomial expansion pro in no time! Now that we've conquered this problem, go forth and conquer more!