Finding The 8th Term Of The Binomial Expansion Of (x+y)^10

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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 invaluable in various fields, including mathematics, statistics, and physics. It allows us to understand the coefficients and terms that arise when we raise a binomial expression to a power. In this comprehensive guide, we will delve into the intricacies of the binomial theorem and, more specifically, how to determine the 8th term in the binomial expansion of (x + y)^10. This involves understanding binomial coefficients, Pascal's triangle, and the general formula for finding a specific term in a binomial expansion. By the end of this discussion, you will have a solid grasp of the underlying principles and the ability to tackle similar problems with confidence.

The binomial theorem is not just a mathematical formula; it is a powerful tool that simplifies complex algebraic expressions. Whether you are a student grappling with algebra, a researcher needing to analyze probabilities, or an engineer solving complex equations, the binomial theorem provides a structured approach to understanding and manipulating binomial expansions. This guide will break down each step, ensuring clarity and comprehension, so you can apply these concepts effectively in your academic and professional pursuits.

Before diving into the specific problem of finding the 8th term, let's quickly recap the binomial theorem itself. The binomial theorem states that for any non-negative integer n:

(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Where:

  • n is the power to which the binomial is raised.
  • k is the term number, starting from 0.
  • (nk)\binom{n}{k} is the binomial coefficient, often read as "n choose k,"

This binomial coefficient represents the number of ways to choose k items from a set of n items and is calculated as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Where:

  • n! denotes the factorial of n, which is the product of all positive integers up to n.

Understanding this formula is the cornerstone of working with binomial expansions. The summation notation indicates that we are summing terms for each value of k from 0 to n. Each term in the expansion consists of three main components: the binomial coefficient, the term a raised to the power of (n-k), and the term b raised to the power of k. Mastering this formula will allow you to expand binomials efficiently and accurately, and it is crucial for solving problems like finding a specific term in an expansion.

In our specific problem, we want to find the 8th term of the binomial expansion (x+y)10(x + y)^{10}. Here, n = 10, and since we start counting terms from k = 0, the 8th term corresponds to k = 7. It's crucial to remember that the term number is one more than the value of k because we begin counting from 0. Misunderstanding this can lead to errors in your calculations.

So, to find the 8th term, we need to calculate the term when k = 7 in the expansion of (x+y)10(x + y)^{10}. This involves substituting n = 10 and k = 7 into the binomial theorem formula. We will need to compute the binomial coefficient (107)\binom{10}{7}, the power of x, and the power of y. This step is the heart of the problem, where the theoretical knowledge of the binomial theorem transforms into practical application. By correctly identifying the values of n and k, we set the stage for a precise calculation of the term we are looking for.

The first step in finding the 8th term is to calculate the binomial coefficient (107)\binom{10}{7}. Using the formula for binomial coefficients, we have:

(107)=10!7!(10−7)!=10!7!3!\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!}

Now, let's break down the factorials:

  • 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
  • 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
  • 3! = 3 × 2 × 1 = 6

Substitute these values back into the equation:

(107)=10×9×8×7!7!×3×2×1\binom{10}{7} = \frac{10 × 9 × 8 × 7!}{7! × 3 × 2 × 1}

Notice that 7! appears in both the numerator and the denominator, so we can cancel it out:

(107)=10×9×83×2×1\binom{10}{7} = \frac{10 × 9 × 8}{3 × 2 × 1}

Now, simplify the expression:

(107)=10×9×86=10×3×4=120\binom{10}{7} = \frac{10 × 9 × 8}{6} = 10 × 3 × 4 = 120

Thus, the binomial coefficient (107)\binom{10}{7} is 120. Calculating binomial coefficients often involves dealing with large factorials, but by recognizing opportunities for simplification, such as canceling out common factors, the computation becomes much more manageable. This calculation is a critical step, as the binomial coefficient serves as the numerical factor in the term we are trying to find. A mistake in this calculation will cascade through the rest of the problem, so precision is key.

Next, we need to determine the powers of x and y in the 8th term. Recall that the general term in the binomial expansion of (x+y)10(x + y)^{10} is given by:

(10k)x10−kyk\binom{10}{k} x^{10-k} y^k

For the 8th term, k = 7. So, we have:

  • The power of x is 10 - k = 10 - 7 = 3
  • The power of y is k = 7

Therefore, in the 8th term, x is raised to the power of 3, and y is raised to the power of 7. Determining these powers is a straightforward application of the binomial theorem formula. The exponent of x decreases as the term number increases, while the exponent of y increases. Understanding this pattern can help you quickly determine the powers without having to memorize the formula. Correctly identifying these powers is crucial because they define the algebraic part of the term, which, when combined with the binomial coefficient, gives us the complete term in the expansion.

Now that we have calculated the binomial coefficient and determined the powers of x and y, we can combine these results to find the 8th term. The 8th term is given by:

(107)x10−7y7=120x3y7\binom{10}{7} x^{10-7} y^7 = 120 x^3 y^7

So, the 8th term in the binomial expansion of (x+y)10(x + y)^{10} is 120x3y7120x^3y^7. This step is the culmination of all the previous calculations. We have successfully computed the binomial coefficient, determined the powers of x and y, and now we combine these to arrive at the final answer. This demonstrates the elegance and efficiency of the binomial theorem, allowing us to directly compute any term in the expansion without having to expand the entire expression.

In summary, the 8th term of the binomial expansion (x+y)10(x + y)^{10} is 120x3y7120x^3y^7. This was determined by understanding and applying the binomial theorem, calculating the binomial coefficient, and finding the appropriate powers of x and y. The binomial theorem is a powerful tool for expanding binomial expressions and is an essential concept in algebra.

By working through this example, you have gained a practical understanding of how to apply the binomial theorem to find a specific term in an expansion. This skill is valuable not only in academic settings but also in various applications that require algebraic manipulation. The ability to confidently handle binomial expansions will undoubtedly enhance your problem-solving capabilities in mathematics and beyond. Remember to practice these techniques to further solidify your understanding and to tackle more complex problems in the future. The binomial theorem, with its structured approach, makes these complex calculations manageable and precise.

Therefore, the correct answer is 120x3y7120 x^3 y^7.