Finding The Ninth Term In The Binomial Expansion Of (x-2y)^13

In the realm of algebra, the binomial theorem stands as a cornerstone for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. This theorem provides a systematic way to determine the coefficients and terms that arise when such binomials are raised to a power. Understanding the binomial theorem is crucial for various mathematical applications, including probability, statistics, and calculus. In this article, we will delve into a specific application of the binomial theorem: finding a particular term in the expansion of a binomial expression. Specifically, we will focus on determining the ninth term in the binomial expansion of $(x - 2y)^{13}$. This exploration will not only reinforce our understanding of the binomial theorem but also highlight the practical steps involved in identifying a specific term within an expansion. By carefully applying the formula and considering the nuances of the given expression, we will arrive at the correct term, illustrating the power and elegance of the binomial theorem.

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. The expansion is a sum of terms, each involving a binomial coefficient, a power of $a$, and a power of $b$. The general formula for the binomial theorem is given by:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

where $\binom{n}{k}$ represents the binomial coefficient, which can be calculated as:

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

Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$. The binomial coefficient $\binom{n}{k}$ gives the number of ways to choose $k$ items from a set of $n$ items, and it plays a crucial role in determining the coefficients in the binomial expansion. In the context of the binomial theorem, $a$ and $b$ can be any algebraic terms, and $n$ is the power to which the binomial is raised. The index $k$ ranges from 0 to $n$, generating each term in the expansion. The term corresponding to $k$ is given by $\binom{n}{k} a^{n-k} b^k$. This formula allows us to expand binomial expressions without having to multiply them out term by term, which can be particularly useful for large values of $n$. Understanding the components of this formula—the binomial coefficient, the powers of $a$ and $b$, and the index $k$—is essential for applying the binomial theorem effectively.

Key Components of the Binomial Theorem

1. Binomial Coefficient: The binomial coefficient, denoted as $\binom{n}{k}$, is a crucial component of the binomial theorem. It represents the number of ways to choose $k$ items from a set of $n$ items without regard to order. The binomial coefficient is calculated using the formula:

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

   where $n!$ (n factorial) is the product of all positive integers up to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. The binomial coefficient appears as the coefficient of each term in the binomial expansion, determining the numerical factor associated with each combination of powers of $a$ and $b$. These coefficients have several interesting properties, such as symmetry ($\binom{n}{k} = \binom{n}{n-k}$) and a recursive relationship that leads to Pascal's Triangle. Pascal's Triangle provides a visual and intuitive way to determine binomial coefficients for small values of $n$, where each number in the triangle is the sum of the two numbers directly above it. Understanding and calculating binomial coefficients is fundamental to applying the binomial theorem, as they dictate the magnitude of each term in the expansion.

2. Powers of a and b: In the binomial expansion of $(a + b)^n$, the powers of a and b play a significant role in determining the structure of each term. The general term in the expansion is given by $\binom{n}{k} a^{n-k} b^k$, where the exponent of $a$ is $n - k$ and the exponent of $b$ is $k$. As $k$ ranges from 0 to $n$, the powers of $a$ decrease from $n$ to 0, while the powers of $b$ increase from 0 to $n$. This inverse relationship between the exponents of $a$ and $b$ ensures that the sum of the exponents in each term is always equal to $n$. For instance, in the expansion of $(x + y)^5$, the terms will have $x$ raised to the powers of 5, 4, 3, 2, 1, and 0, while $y$ will be raised to the powers of 0, 1, 2, 3, 4, and 5, respectively. This systematic variation of powers allows for the complete expansion of the binomial expression, capturing all possible combinations of $a$ and $b$. The correct application of these exponents is crucial for obtaining the accurate terms in the expansion.

3. Index k: The index $k$ in the binomial theorem serves as a counter that determines the position and composition of each term in the expansion of $(a + b)^n$. In the general term $\binom{n}{k} a^{n-k} b^k$, the index $k$ ranges from 0 to $n$, generating each term in the expansion. When $k = 0$, we obtain the first term, and when $k = n$, we obtain the last term. The value of $k$ directly affects the powers of $a$ and $b$, as well as the binomial coefficient. Specifically, the exponent of $b$ is equal to $k$, and the exponent of $a$ is $n - k$. For example, if we want to find the third term in the expansion, we set $k = 2$ (since the first term corresponds to $k = 0$, the second to $k = 1$, and so on). The binomial coefficient $\binom{n}{k}$ also depends on $k$, influencing the numerical factor of the term. Understanding the role of the index $k$ is essential for identifying a specific term in the binomial expansion, as it allows us to systematically navigate through the terms and pinpoint the one we are interested in. By correctly determining the value of $k$ that corresponds to the desired term, we can efficiently apply the binomial theorem to find the term without expanding the entire expression.

Problem Statement: Finding the Ninth Term

Our main objective is to identify the ninth term in the binomial expansion of $(x - 2y)^{13}$. To achieve this, we will employ the binomial theorem, which provides a structured approach for expanding expressions of this form. The binomial theorem states that for any non-negative integer $n$:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

In our case, $a = x$, $b = -2y$, and $n = 13$. The index $k$ starts at 0 for the first term, 1 for the second term, and so on. Therefore, to find the ninth term, we need to determine the term corresponding to $k = 8$ (since the first term corresponds to $k = 0$, the second to $k = 1$, and so on). By substituting these values into the general formula, we can isolate and calculate the ninth term. This involves computing the binomial coefficient $\binom{13}{8}$, determining the powers of $x$ and $-2y$, and combining these elements to form the specific term. The process requires careful attention to detail, particularly in handling the negative sign in $-2y$ and ensuring the correct exponents are applied. By systematically applying the binomial theorem, we can efficiently find the ninth term without needing to expand the entire expression. This approach demonstrates the power and utility of the binomial theorem in solving algebraic problems involving binomial expansions.

Applying the Binomial Theorem to Find the Ninth Term

To find the ninth term in the binomial expansion of $(x - 2y)^{13}$, we use the binomial theorem. The general term in the expansion is given by:

$$\binom{n}{k} a^{n-k} b^k$$

In our problem, $a = x$, $b = -2y$, and $n = 13$. Since we are looking for the ninth term, we need to find the term corresponding to $k = 8$. This is because the first term corresponds to $k = 0$, the second to $k = 1$, and so on.

Substituting these values into the formula, we get:

$$\binom{13}{8} x^{13-8} (-2y)^8$$

First, we calculate the binomial coefficient $\binom{13}{8}$:

$$\binom{13}{8} = \frac{13!}{8!(13-8)!} = \frac{13!}{8!5!} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 13 \times 11 \times 9 = 1287$$

Next, we simplify the powers of $x$ and $-2y$:

$$x^{13-8} = x^5$$

$$(-2y)^8 = (-2)^8 y^8 = 256 y^8$$

Now, we combine these results:

$$1287 imes x^5 imes 256 y^8 = 329472 x^5 y^8$$

Thus, the ninth term in the binomial expansion of $(x - 2y)^{13}$ is $329,472 x^5 y^8$. This methodical application of the binomial theorem allows us to efficiently pinpoint the desired term without needing to expand the entire expression, showcasing the theorem's practical utility in algebraic manipulations.

Step-by-Step Calculation

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

   • $n = 13$ (the exponent of the binomial)
   • $a = x$ (the first term in the binomial)
   • $b = -2y$ (the second term in the binomial)
   • $k = 8$ (since we want the ninth term, and the count starts from 0) Identifying the correct values for $n$, $a$, $b$, and $k$ is the foundational step in applying the binomial theorem to find a specific term in the expansion. The exponent $n$ dictates the overall power of the binomial expression, while $a$ and $b$ are the individual terms within the binomial. The value of $k$ is crucial as it determines the position of the term we are seeking in the expansion, with $k = 0$ corresponding to the first term, $k = 1$ to the second, and so on. In this case, recognizing that the ninth term corresponds to $k = 8$ is essential for the subsequent calculations. A clear understanding of these variables ensures that the binomial theorem is applied correctly, leading to an accurate determination of the desired term. This initial step sets the stage for the rest of the calculation, providing a structured framework for applying the formula.

2. Calculate the binomial coefficient:

   $$\binom{13}{8} = \frac{13!}{8!(13-8)!} = \frac{13!}{8!5!} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287$$

   Calculating the binomial coefficient is a critical step in determining the magnitude of a particular term in the binomial expansion. The binomial coefficient, denoted as $\binom{n}{k}$, represents the number of ways to choose $k$ items from a set of $n$ items, and it is computed using the formula $\frac{n!}{k!(n-k)!}$. In our case, we need to calculate $\binom{13}{8}$, which involves dividing the factorial of 13 by the product of the factorials of 8 and 5. This calculation can be simplified by canceling out common factors in the numerator and the denominator, making the computation more manageable. The result, 1287, represents the numerical coefficient associated with the ninth term in the binomial expansion of $(x - 2y)^{13}$. This coefficient is crucial for obtaining the accurate term, as it scales the product of the powers of $x$ and $-2y$. Accurate calculation of the binomial coefficient is thus essential for the correct application of the binomial theorem.

3. Determine the powers of x and -2y:

   • $x^{13-8} = x^5$
   • $(-2y)^8 = (-2)^8 y^8 = 256 y^8$ Determining the powers of x and -2y involves applying the exponents in the binomial theorem formula to the specific terms within our binomial expression. According to the binomial theorem, the power of $a$ (in this case, $x$) is given by $n - k$, and the power of $b$ (in this case, $-2y$) is given by $k$. For the ninth term, where $k = 8$ and $n = 13$, the power of $x$ is $13 - 8 = 5$, resulting in $x^5$. For the term $-2y$, the power is $8$, which means we need to calculate $(-2y)^8$. This involves raising both the coefficient -2 and the variable $y$ to the power of 8. Since $(-2)^8 = 256$, we get $(-2y)^8 = 256y^8$. The correct determination of these powers is crucial for constructing the specific term in the binomial expansion, as these exponents dictate the algebraic structure of the term. A careful application of the power rules ensures that the resulting term accurately reflects the contribution of $x$ and $-2y$ in the expansion.

4. Multiply the binomial coefficient with the powers of x and -2y:

   $$1287 imes x^5 imes 256 y^8 = 329472 x^5 y^8$$

   Multiplying the binomial coefficient with the powers of $x$ and $-2y$ is the final step in determining the complete expression for the ninth term in the binomial expansion. This step combines the numerical coefficient, derived from the binomial coefficient, with the algebraic components, which are the powers of $x$ and $-2y$. In this case, we multiply the binomial coefficient 1287 with $x^5$ and $256y^8$. This involves multiplying the numerical values 1287 and 256, which yields 329472. We then combine this numerical result with the variable terms $x^5$ and $y^8$ to obtain the final term: $329472 x^5 y^8$. This resulting term represents the ninth term in the binomial expansion of $(x - 2y)^{13}$. The accurate execution of this multiplication ensures that the term is correctly represented with its appropriate numerical and algebraic components, completing the process of finding the desired term using the binomial theorem.

Conclusion

In conclusion, we have successfully determined the ninth term in the binomial expansion of $(x - 2y)^{13}$ by systematically applying the binomial theorem. The process involved identifying the key components, such as the binomial coefficient, the powers of the terms in the binomial, and the correct index value corresponding to the ninth term. We calculated the binomial coefficient $\binom{13}{8}$ as 1287, determined the powers of $x$ and $-2y$ as $x^5$ and $256y^8$ respectively, and then multiplied these components together to obtain the ninth term: $329,472 x^5 y^8$. This methodical approach highlights the power and utility of the binomial theorem in efficiently finding specific terms in binomial expansions without needing to expand the entire expression. Understanding and applying the binomial theorem is a fundamental skill in algebra, with applications in various fields of mathematics and beyond. The step-by-step calculation presented in this article provides a clear and concise guide for solving similar problems, reinforcing the importance of each component in the formula and the overall process of binomial expansion.