Binomial Expansion: Finding The Fourth Term Of (e + 2f)^10

by Alex Johnson 59 views

Understanding binomial expansions can seem daunting, but breaking down the process into manageable steps makes it much more approachable. In this article, we'll focus on finding a specific term within a binomial expansion. Specifically, we'll tackle the question: What is the fourth term in the binomial expansion of (e + 2f)^10? To answer this question, we will delve into the binomial theorem, its components, and how to apply it effectively.

Understanding the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. This theorem is a cornerstone of algebra and calculus, offering a systematic way to understand polynomial expansions. It states that:

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

where the summation (Σ) runs from k = 0 to n, and (n choose k) represents the binomial coefficient, also written as nCk or binom(n, k). The binomial coefficient is calculated as:

(n choose k) = n! / (k! * (n-k)!)

where "!" denotes the factorial function. The factorial of a non-negative integer m, denoted by m!, is the product of all positive integers less than or equal to m. For instance, 5! = 5 × 4 × 3 × 2 × 1 = 120. Understanding these notations and formulas is crucial for navigating binomial expansions. The beauty of the binomial theorem lies in its ability to predict the coefficients and terms of an expanded binomial without having to manually multiply the expression multiple times. This not only saves time but also reduces the likelihood of errors.

The binomial theorem is widely used in various fields, including probability, statistics, and computer science. In probability theory, binomial coefficients appear in the context of binomial distributions, which describe the probability of a certain number of successes in a sequence of independent trials. In computer science, they are used in algorithms related to combinations and permutations. Therefore, mastering the binomial theorem has implications beyond pure mathematics, extending into practical applications in diverse areas.

Key Components of the Binomial Theorem

Let's dissect the key components of the binomial theorem to better understand its mechanics:

  1. Binomial Coefficient (n choose k): This coefficient represents the number of ways to choose k elements from a set of n elements. It determines the numerical factor in each term of the expansion. For example, (10 choose 3) tells us how many ways we can select 3 items from a set of 10. This concept is fundamental in combinatorics and probability, providing a mathematical basis for counting combinations and permutations.

  2. Terms a^(n-k) and b^k: These represent the powers of the terms 'a' and 'b' in the binomial (a + b). Notice that the exponents (n-k) and k add up to n, the original power of the binomial. As k increases from 0 to n, the exponent of 'a' decreases while the exponent of 'b' increases. This systematic change in exponents is what generates the sequence of terms in the expansion. Understanding this pattern allows us to predict the structure of the expansion and identify specific terms without having to calculate all preceding terms.

  3. Summation (Σ): The summation symbol indicates that we are summing terms for each value of k, starting from k = 0 and going up to k = n. Each value of k corresponds to a different term in the expansion. The summation notation provides a concise way to represent the entire expansion, encapsulating all the individual terms within a single expression. This makes it easier to work with the binomial theorem in mathematical proofs and calculations.

By understanding these components, we can appreciate how the binomial theorem works and how to apply it effectively to various problems. Now, let's apply this knowledge to find the fourth term in the expansion of (e + 2f)^10.

Applying the Binomial Theorem to Find the Fourth Term

Now, let's apply the binomial theorem to our specific problem: finding the fourth term in the expansion of (e + 2f)^10. The binomial theorem formula is:

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

In our case, a = e, b = 2f, and n = 10. Since we are looking for the fourth term, we need to consider that the terms are numbered starting from 0. Therefore, the fourth term corresponds to k = 3 (since the first term is k = 0, the second is k = 1, the third is k = 2, and the fourth is k = 3).

Thus, we need to calculate the term when k = 3. Plugging in the values, we get:

Fourth term = (10 choose 3) * e^(10-3) * (2f)^3

Now, let's calculate each part:

  1. (10 choose 3): This is the binomial coefficient, which we calculate as:

    (10 choose 3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120

  2. e^(10-3): This simplifies to e^7.

  3. (2f)^3: This simplifies to 8f^3 (since 2^3 = 8).

Putting it all together, the fourth term is:

Fourth term = 120 * e^7 * 8f^3 = 960e7f3

Therefore, the fourth term in the binomial expansion of (e + 2f)^10 is 960e7f3. This step-by-step calculation illustrates the power of the binomial theorem in isolating specific terms within an expansion, saving us from having to compute the entire series. The key is to correctly identify the values of a, b, n, and k, and then apply the formula methodically.

Step-by-Step Calculation

To solidify understanding, let's reiterate the step-by-step calculation:

  1. Identify n, a, b, and k:

    • n = 10
    • a = e
    • b = 2f
    • k = 3 (for the fourth term)

  2. Calculate the binomial coefficient (n choose k):

    • (10 choose 3) = 10! / (3! * 7!) = 120

  3. Determine a^(n-k):

    • e^(10-3) = e^7

  4. Determine b^k:

    • (2f)^3 = 8f^3

  5. Multiply the components together:

    • Fourth term = 120 * e^7 * 8f^3 = 960e7f3

This structured approach not only helps in accurately calculating the terms but also reinforces the understanding of the underlying principles of the binomial theorem. By breaking the problem down into smaller steps, we can tackle complex expansions with greater confidence and accuracy. Each step serves a specific purpose, contributing to the final result in a clear and logical manner.

Common Mistakes to Avoid

When working with binomial expansions, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and improve your accuracy.

  1. Incorrectly Identifying the Term Number: A frequent mistake is misidentifying the term number. Remember that the terms in a binomial expansion start from k = 0. So, the fourth term corresponds to k = 3, not k = 4. This off-by-one error can throw off the entire calculation. Always double-check the term number you are trying to find and adjust the value of k accordingly. It’s a simple oversight, but one that can easily be prevented with careful attention to detail.

  2. Miscalculating the Binomial Coefficient: The binomial coefficient calculation can be tricky, especially with larger numbers. Make sure to correctly apply the formula n! / (k! * (n-k)!). Errors often occur in factorial calculations or in the division. Using a calculator with a factorial function or a binomial coefficient function can help reduce these errors. Alternatively, writing out the factorials and simplifying terms can make the calculation more manageable. Precision in this step is vital, as the binomial coefficient is a multiplicative factor in the final term.

  3. Forgetting to Apply the Exponent to the Entire Term: When dealing with a term like (2f)^3, it's essential to apply the exponent to both the constant and the variable. Many students mistakenly calculate this as 2f^3 instead of 8f^3. Pay close attention to the parentheses and ensure that the exponent is distributed correctly. This is a common algebraic error that can easily be overlooked if one isn't careful.

  4. Arithmetic Errors: Simple arithmetic mistakes in the final multiplication can also lead to incorrect answers. Double-check your calculations, especially when multiplying large numbers or dealing with exponents. Using a calculator and taking your time can help minimize these errors. It’s always a good practice to review the entire calculation from start to finish to catch any potential slips.

By being mindful of these common errors and taking steps to avoid them, you can significantly improve your accuracy when working with binomial expansions.

Conclusion

In summary, finding the fourth term in the binomial expansion of (e + 2f)^10 involves understanding and applying the binomial theorem. By correctly identifying the values of n, a, b, and k, calculating the binomial coefficient, and simplifying the terms, we arrived at the solution: 960e7f3. Avoiding common mistakes and practicing these steps will enhance your ability to tackle binomial expansion problems confidently. The binomial theorem is a powerful tool in mathematics, and mastering its application can be beneficial in various fields.

For further exploration and a deeper understanding of binomial expansions, consider visiting Khan Academy's Binomial Theorem section.