Binomial Theorem: Finding The Third Term Of (x-3)^9

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Hey guys! Today, we're diving into the fascinating world of the Binomial Theorem. Specifically, we're going to tackle the question of how to find a particular term in a binomial expansion. Our focus will be on finding the third term in the expansion of the expression (x-3)^9. So, buckle up, and let's get started!

Understanding the Binomial Theorem

Before we jump into the problem, let's quickly recap what the Binomial Theorem actually is. The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where 'n' is a non-negative integer. It's a powerful tool that allows us to avoid tedious manual multiplication when dealing with higher powers. The general formula for the Binomial Theorem is:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

Where:

  • 'Σ' represents the summation from k = 0 to n.
  • '(n choose k)' is the binomial coefficient, which can be calculated as n! / (k! * (n-k)!).
  • 'n!' denotes the factorial of n (e.g., 5! = 5 * 4 * 3 * 2 * 1).

This formula might look a bit intimidating at first, but don't worry! We'll break it down step by step as we apply it to our specific problem. Understanding each part of the formula is key to successfully using the Binomial Theorem. The binomial coefficient, in particular, tells us the numerical coefficient of each term in the expansion. Think of it as a way to count the number of ways to choose 'k' items from a set of 'n' items. This is a crucial concept in combinatorics and probability, and it plays a central role in the Binomial Theorem. Knowing how to calculate factorials and binomial coefficients is essential for working with this theorem effectively.

Moreover, it's important to note that the Binomial Theorem is not just a mathematical curiosity; it has practical applications in various fields, including statistics, probability, and computer science. For example, it can be used to approximate probabilities in situations where calculating them directly would be difficult. In computer science, it finds applications in algorithms and data structures. So, understanding the Binomial Theorem is not only useful for solving mathematical problems but also for building a foundation for more advanced concepts in other areas.

Identifying the Third Term

Now that we have a handle on the Binomial Theorem, let's focus on our main task: finding the third term in the expansion of (x - 3)^9. A crucial thing to remember is that the terms in the expansion are numbered starting from 0. This means:

  • The first term corresponds to k = 0.
  • The second term corresponds to k = 1.
  • Therefore, the third term corresponds to k = 2.

This might seem a bit counterintuitive at first, but it's a standard convention in mathematics when dealing with series and sequences. So, to find the third term, we'll plug k = 2 into the Binomial Theorem formula. It's a common mistake to assume that the third term corresponds to k = 3, so always double-check this! Understanding this indexing is vital to correctly applying the formula and arriving at the right answer. If you get the 'k' value wrong, the entire calculation will be off. So, remember, we're looking for the term where k = 2.

Another way to think about this is to visualize the expansion process. When you expand a binomial like (a + b)^n, you're essentially multiplying (a + b) by itself 'n' times. Each term in the expansion arises from a specific combination of choosing either 'a' or 'b' from each of these 'n' factors. The value of 'k' tells you how many times you've chosen 'b'. So, for the third term (k = 2), you're looking at the combinations where you've chosen 'b' exactly twice. This perspective can help you develop a more intuitive understanding of why the indexing starts from 0 and how the Binomial Theorem works.

Applying the Formula

With k = 2 identified, we can now plug the values into the Binomial Theorem formula. In our case, a = x, b = -3, and n = 9. So, the third term will be:

(9 choose 2) * x^(9-2) * (-3)^2

Let's break this down step by step:

  1. (9 choose 2): This is the binomial coefficient, which we calculate as 9! / (2! * 7!).
  2. x^(9-2): This simplifies to x^7.
  3. (-3)^2: This simplifies to 9.

Now, let's calculate the binomial coefficient. Remember, 9! means 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, and similarly for 2! and 7!. We can simplify the calculation by canceling out the 7! term in the numerator and denominator:

(9 choose 2) = 9! / (2! * 7!) = (9 * 8 * 7!) / (2 * 1 * 7!) = (9 * 8) / 2 = 36

So, the binomial coefficient (9 choose 2) is 36. Now we have all the pieces to put together the third term.

It's crucial to be careful with the negative sign in 'b = -3'. When raising a negative number to an even power, the result is positive, but when raising it to an odd power, the result is negative. This is a common source of errors when applying the Binomial Theorem. Pay close attention to the exponent of 'b' to ensure you get the correct sign. Also, remember that the order of operations matters. You need to calculate the binomial coefficient and the powers of 'a' and 'b' before multiplying them together. This methodical approach will help you avoid mistakes and arrive at the correct answer. Practicing more examples will help you become more comfortable with the formula and the calculations involved.

Calculating the Third Term

Now we can substitute the values we calculated back into the expression for the third term:

Third Term = 36 * x^7 * 9

Multiplying the constants, we get:

Third Term = 324x^7

And there you have it! The third term in the expansion of (x - 3)^9 is 324x^7.

This result is a single term, a combination of the coefficient (324) and the variable part (x^7). It represents one specific piece of the entire expansion of (x - 3)^9. If you were to expand the entire expression, you would get a polynomial with 10 terms (since the power is 9), and 324x^7 would be just one of them. Understanding how to find a specific term without expanding the entire binomial is a key advantage of using the Binomial Theorem. It saves you a lot of time and effort, especially when dealing with higher powers.

Moreover, it's worth noting that the coefficients in the binomial expansion follow a pattern known as Pascal's Triangle. Each number in Pascal's Triangle is the sum of the two numbers directly above it. The rows of Pascal's Triangle correspond to the binomial coefficients for different powers of (a + b). While we used the formula to calculate the binomial coefficient in this case, you could also use Pascal's Triangle to find it, especially for smaller values of 'n'.

Conclusion

So, guys, we've successfully used the Binomial Theorem to find the third term in the expansion of (x - 3)^9. We've seen how to identify the correct value of 'k', apply the formula, and perform the necessary calculations. Remember, the key to mastering the Binomial Theorem is understanding the formula and practicing applying it to different problems. Keep practicing, and you'll become a pro in no time!

The Binomial Theorem is a powerful tool in mathematics, and understanding it will open doors to more advanced concepts. It's not just about memorizing the formula; it's about understanding the underlying principles and how they connect to other areas of mathematics. So, keep exploring, keep questioning, and keep learning! You've got this!