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

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 especially useful when dealing with high powers, as it allows us to avoid the tedious process of repeatedly multiplying the binomial by itself. At its core, the binomial theorem tells us how to expand expressions like $(x - 2y)^{13}$ into a sum of terms involving powers of $x$ and $y$. Each term in the expansion has a coefficient and a specific combination of powers of $x$ and $y$. The binomial theorem provides a formula to calculate these coefficients and powers directly, without having to manually multiply out the expression.

The general form of the binomial theorem is given by:

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

Here, $\binom{n}{k}$ represents the binomial coefficient, which is often read as "n choose k" and is calculated as:

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

Where $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. The binomial coefficient $\binom{n}{k}$ gives the coefficient of the term $a^{n-k} b^k$ in the expansion of $(a + b)^n$. Understanding this formula is crucial for finding specific terms in a binomial expansion, as it allows us to determine the coefficient and the powers of each variable without expanding the entire expression.

In the context of our problem, we need to find the ninth term in the expansion of $(x - 2y)^{13}$. The binomial theorem provides us with the tools to do this efficiently. By identifying the values of $a$, $b$, and $n$, and then applying the formula, we can find the desired term without having to calculate all the preceding terms. This is particularly useful when dealing with large powers, where manual expansion would be extremely time-consuming and error-prone. Therefore, mastering the binomial theorem is essential for solving problems involving binomial expansions and is a cornerstone of algebraic manipulation.

To apply the binomial theorem effectively, we must first identify the components in our given expression, $(x - 2y)^{13}$. The binomial theorem, as stated earlier, is $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. Comparing this with our expression, we can identify the following:

• $a = x$
• $b = -2y$
• $n = 13$

Here, $a$ corresponds to the first term in the binomial, which is $x$. The term $b$ corresponds to the second term, which is $-2y$. It is crucial to include the negative sign in this case because it will affect the sign of the terms in the expansion. The exponent $n$ is 13, which indicates that we are raising the binomial to the power of 13. Identifying these components correctly is the first step in using the binomial theorem to find a specific term in the expansion. Incorrectly identifying these components will lead to errors in the subsequent calculations.

Now, we need to determine which value of $k$ corresponds to the ninth term in the expansion. It's important to note that the binomial theorem starts counting terms from $k = 0$. This means that:

• The first term corresponds to $k = 0$
• The second term corresponds to $k = 1$
• The third term corresponds to $k = 2$
• And so on...

Following this pattern, the ninth term corresponds to $k = 8$. This is because we start counting from 0, so the ninth term is reached when $k$ is 8. Understanding this indexing is crucial for correctly applying the binomial theorem to find the desired term. If we were looking for the third term, we would use $k = 2$; for the fifth term, $k = 4$; and so on. This indexing convention is a common source of confusion, so it's essential to keep it in mind when solving problems involving the binomial theorem. By correctly identifying $k$, we can plug the values into the binomial theorem formula and calculate the exact term we are looking for.

Now that we have identified the components $a = x$, $b = -2y$, $n = 13$, and $k = 8$, we can proceed to calculate the ninth term in the expansion of $(x - 2y)^{13}$ using the binomial theorem. The general term in the binomial expansion is given by:

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

Substituting the values we identified, we get:

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

First, let's calculate the binomial coefficient $\binom{13}{8}$. Recall that the binomial coefficient is calculated as:

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

So, for our case, we have:

$\binom{13}{8} = \frac{13!}{8!(13-8)!} = \frac{13!}{8!5!}$

Now, we compute the factorials:

$13! = 13 \times 12 \times 11 \times 10 \times 9 \times 8!$

$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Substitute these values into the binomial coefficient formula:

$\binom{13}{8} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8!}{8! \times 5!}$

We can cancel out $8!$ from the numerator and denominator:

$\binom{13}{8} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5!}$

Now, substitute $5! = 120$:

$\binom{13}{8} = \frac{13 \times 12 \times 11 \times 10 \times 9}{120}$

Simplify the expression:

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

So, the binomial coefficient $\binom{13}{8}$ is 1287. Next, we need to calculate the powers of $x$ and $y$:

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

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

Now, substitute these values back into the term formula:

$\binom{13}{8} x^{13-8} (-2y)^8 = 1287 x^5 (256 y^8)$

Multiply the coefficient:

$1287 \times 256 = 329,472$

Therefore, the ninth term in the binomial expansion is:

$329,472 x^5 y^8$

In conclusion, to find the ninth term in the binomial expansion of $(x - 2y)^{13}$, we applied the binomial theorem. We identified the components $a = x$, $b = -2y$, and $n = 13$. We then determined that the ninth term corresponds to $k = 8$ in the binomial expansion formula. By substituting these values into the formula $\binom{n}{k} a^{n-k} b^k$, we calculated the binomial coefficient $\binom{13}{8}$ as 1287.

Next, we computed the powers of $x$ and $y$ as $x^{13-8} = x^5$ and $(-2y)^8 = 256y^8$. Multiplying these values together, we obtained the ninth term as $1287 \times x^5 \times 256y^8$. Finally, we multiplied the coefficients to get $329,472 x^5 y^8$. Thus, the ninth term in the binomial expansion of $(x - 2y)^{13}$ is $329,472 x^5 y^8$.

Therefore, the correct answer is:

A. $329,472 x^5 y^8$

This process demonstrates the power and efficiency of the binomial theorem in finding specific terms in a binomial expansion without having to expand the entire expression. By carefully identifying the components, applying the formula, and performing the calculations step-by-step, we can accurately determine any term in the expansion. This method is particularly useful for higher powers, where manual expansion would be impractical. Understanding and applying the binomial theorem is a crucial skill in algebra and provides a foundation for more advanced mathematical concepts.