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

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The binomial theorem is a fundamental concept in algebra that provides a method for expanding expressions of the form (a + b)^n, where n is a non-negative integer. This theorem is crucial in various fields, including mathematics, statistics, and physics. In this comprehensive article, we will delve into the binomial theorem, its applications, and, most importantly, how to find a specific term in a binomial expansion. We will focus on determining the second term in the binomial expansion of (2r - 3s)^12, offering a step-by-step solution and an in-depth explanation.

The Binomial Theorem: A Quick Overview

At its core, the binomial theorem provides a formula for expanding expressions like (a + b)^n. The general formula is:

(a + b)^n = Σ [nCk * a^(n-k) * b^k]

where:

  • n is a non-negative integer (the power to which the binomial is raised).
  • k is an integer ranging from 0 to n.
  • nCk represents the binomial coefficient, also written as "n choose k", which is calculated as n! / (k!(n-k)!). The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.

This formula tells us that the expansion of (a + b)^n will consist of (n + 1) terms. Each term will have the form of a binomial coefficient multiplied by powers of 'a' and 'b'. The exponents of 'a' decrease from n to 0, while the exponents of 'b' increase from 0 to n.

Understanding Binomial Coefficients

Binomial coefficients, often denoted as nCk or "n choose k", play a crucial role in the binomial theorem. They represent the number of ways to choose k elements from a set of n elements without regard to order. The formula for calculating binomial coefficients is:

nCk = n! / (k!(n-k)!)

For example, if we want to calculate 5C2 (5 choose 2), we use the formula:

5C2 = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 * 4 * 3 * 2 * 1) / ((2 * 1)(3 * 2 * 1)) = 120 / (2 * 6) = 10

This means there are 10 ways to choose 2 elements from a set of 5 elements. Binomial coefficients can also be found using Pascal's Triangle, which provides a visual representation of these coefficients.

Pascal's Triangle and Binomial Coefficients

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The rows of Pascal's Triangle correspond to the binomial coefficients for different values of n. The top row (row 0) corresponds to n = 0, the second row (row 1) corresponds to n = 1, and so on.

Each number in Pascal's Triangle is a binomial coefficient. The kth element in the nth row (starting from k = 0) is nCk. For example, the 4th row (n = 3) of Pascal's Triangle is 1 3 3 1, which corresponds to the coefficients 3C0, 3C1, 3C2, and 3C3. Using Pascal's Triangle can be a quick way to find binomial coefficients for small values of n.

Applications of the Binomial Theorem

The binomial theorem has numerous applications across various fields. Some key applications include:

  1. Probability Theory: The binomial theorem is used to calculate probabilities in binomial distributions, which model the probability of a certain number of successes in a fixed number of independent trials.
  2. Combinatorics: It is used in counting problems and combinatorial analysis to determine the number of ways to choose items from a set.
  3. Calculus: The binomial theorem can be used to approximate functions and evaluate limits.
  4. Computer Science: It is used in algorithms for polynomial multiplication and other algebraic computations.
  5. Physics: The binomial theorem appears in various physical calculations, such as in quantum mechanics and statistical mechanics.

Understanding the binomial theorem is crucial for solving problems in these areas and many others.

Finding a Specific Term in a Binomial Expansion

Now, let's focus on how to find a specific term in a binomial expansion. The general formula for the (k+1)th term in the expansion of (a + b)^n is:

Tk+1 = nCk * a^(n-k) * b^k

Here, k starts from 0, so the first term corresponds to k = 0, the second term to k = 1, the third term to k = 2, and so on. To find the second term, we set k = 1.

Step-by-Step Solution for the Second Term of (2r - 3s)^12

We are tasked with finding the second term in the binomial expansion of (2r - 3s)^12. To do this, we follow these steps:

  1. Identify n, a, and b: In our expression, (2r - 3s)^12, we have n = 12, a = 2r, and b = -3s.
  2. Determine k for the second term: Since we want the second term, k = 1 (as the first term corresponds to k = 0).
  3. Apply the formula for the (k+1)th term: We use the formula Tk+1 = nCk * a^(n-k) * b^k with n = 12, k = 1, a = 2r, and b = -3s.

T2 = 12C1 * (2r)^(12-1) * (-3s)^1

  1. Calculate the binomial coefficient: Calculate 12C1 using the formula nCk = n! / (k!(n-k)!):

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

  1. Calculate the powers of a and b: Calculate (2r)^(12-1) and (-3s)^1:

(2r)^11 = 2^11 * r^11 = 2048r^11

(-3s)^1 = -3s

  1. Substitute the values into the formula: Plug the calculated values back into the formula for T2:

T2 = 12 * 2048r^11 * (-3s)

  1. Simplify the expression: Multiply the coefficients and combine the terms:

T2 = 12 * 2048 * (-3) * r^11 * s = -73728r^11s

Therefore, the second term in the binomial expansion of (2r - 3s)^12 is -73,728r^11s.

Common Mistakes to Avoid

When working with the binomial theorem, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and solve problems more accurately.

  1. Incorrectly identifying a and b: Make sure to correctly identify the terms 'a' and 'b' in the binomial expression. Pay close attention to the signs. For example, in the expression (2r - 3s)^12, 'a' is 2r and 'b' is -3s, not 3s.
  2. Using the wrong value for k: Remember that the (k+1)th term corresponds to the index k. If you want the second term, use k = 1, not k = 2. This is a frequent source of error.
  3. Miscalculating the binomial coefficient: Ensure you calculate nCk correctly using the formula n! / (k!(n-k)!). Mistakes in calculating factorials or dividing can lead to incorrect results.
  4. Forgetting to apply the exponent to the coefficient: When raising a term like (2r) to a power, remember to apply the exponent to both the coefficient and the variable. For example, (2r)^11 is 2^11 * r^11, not just 2 * r^11.
  5. Incorrectly handling negative signs: Pay close attention to negative signs, especially when raising negative numbers to a power. A negative number raised to an odd power will be negative, while a negative number raised to an even power will be positive.

By avoiding these common mistakes, you can improve your accuracy and confidence in applying the binomial theorem.

Conclusion

The binomial theorem is a powerful tool for expanding expressions of the form (a + b)^n and has wide-ranging applications in various fields. Understanding the theorem and its applications is essential for success in algebra and beyond. In this article, we have explored the binomial theorem, discussed how to calculate binomial coefficients, and provided a step-by-step solution for finding the second term in the binomial expansion of (2r - 3s)^12. The correct answer is C. -73,728r^11s. By following the outlined steps and avoiding common mistakes, you can confidently tackle similar problems and deepen your understanding of this important mathematical concept. Whether you are studying for an exam or applying the binomial theorem in real-world scenarios, a solid grasp of this theorem will serve you well.