Find The Coefficient Of The $x^9y$ Term In A Binomial Expansion

Hey everyone! Let's crack this math problem together. We're diving into the world of binomial expansion to find the coefficient of a specific term. Specifically, we're looking for the coefficient of the $x^9y$ term in the expansion of $\left(2y + 4x^3\right)^4$. Sounds fun, right? Don't worry, it's not as scary as it seems. We'll break it down step by step, making sure everyone understands the process. This isn't just about getting an answer; it's about understanding the why behind the math. Are you ready to unravel this mathematical mystery? Let's go!

Understanding the Binomial Theorem: The Foundation

So, before we jump into the problem, let's quickly recap the binomial theorem. It's like the secret weapon for expanding expressions like the one we have. The binomial theorem gives us a formula to expand expressions of the form $(a + b)^n$. The general form of the expansion is:

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

Where:

• $n$ is a non-negative integer (in our case, 4).
• $k$ is an integer from 0 to $n$ (0, 1, 2, 3, and 4 in our case).
• $\binom{n}{k}$ is the binomial coefficient, also written as $C(n, k)$ or $_nC_k$, which is calculated as $\frac{n!}{k!(n-k)!}$. This tells us how many ways we can choose $k$ items from a set of $n$ items.
• $a$ and $b$ are the terms within the binomial (in our case, $2y$ and $4x^3$).

Essentially, the binomial theorem allows us to expand a binomial expression raised to a power into a sum of terms, each with a specific coefficient and powers of $a$ and $b$. Think of it as a structured way to multiply out the expression, without actually having to do the repeated multiplication. Pretty neat, huh? Understanding the binomial theorem is like having the keys to unlock this type of problem. With this in mind, we can simplify the main question and make it easier to tackle.

Applying the Binomial Theorem to Our Problem

Alright, let's apply the binomial theorem to our specific expression: $\left(2y + 4x^3\right)^4$. In this case:

• $a = 2y$
• $b = 4x^3$
• $n = 4$

We want to find the term with $x^9y$. The general term in the expansion will be:

$\binom{4}{k} (2y)^{4-k} (4x^3)^k$

To get the $x^9$ term, we need the power of $x$ from $(4x^3)^k$ to be 9. So, we need to find the value of $k$ such that $3k = 9$. Solving for $k$, we get $k = 3$. Therefore, we can substitute $k=3$ in the general term to find the desired term:

$\binom{4}{3} (2y)^{4-3} (4x^3)^3$

Let's break this down step by step. First, let's calculate the binomial coefficient: $\binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(1)} = 4$. Then, we need to find the coefficients of each term. So, we have $(2y)^{4-3} = (2y)^1 = 2y$ and $(4x^3)^3 = 4^3 * (x^3)^3 = 64x^9$. Now, we can multiply it all together to get $4 * (2y) * (64x^9) = 512yx^9$.

Calculating the Specific Term

Now that we've identified the correct term, let's calculate it precisely. We know that $k = 3$. So, we plug this into our general term formula:

Term = $\binom{4}{3} (2y)^{4-3} (4x^3)^3$

Let's simplify this:

• $\binom{4}{3} = 4$
• $(2y)^{4-3} = (2y)^1 = 2y$
• $(4x^3)^3 = 4^3 * (x^3)^3 = 64x^9$

Putting it all together:

Term = $4 \times 2y \times 64x^9 = 512x^9y$

So, the term we're looking for is $512x^9y$. The coefficient of the $x^9y$ term is 512.

The Final Answer

Therefore, the coefficient of the $x^9y$ term in the expansion of $\left(2y + 4x^3\right)^4$ is 512. The answer is D. We used the binomial theorem to systematically expand the expression and then identified the term that matched our criteria ($x^9y$). By carefully calculating the binomial coefficients and the powers of $x$ and $y$, we were able to arrive at the correct answer. Awesome, right? The process involves recognizing the binomial theorem, applying it to the specific problem, and then carefully simplifying the expression to get the desired term. This problem highlights the importance of understanding the underlying mathematical principles and how to apply them step-by-step. Congratulations on reaching the end and understanding the coefficient.

Tips for Solving Similar Problems

Here are some handy tips to help you conquer similar problems in the future:

• Master the Binomial Theorem: Make sure you understand the formula and how to use it. Practice expanding different binomials to build your confidence.
• Identify $a$, $b$, and $n$: Carefully identify the components of the binomial expression. This is the first and most important step.
• Determine the Value of $k$: Figure out the value of $k$ that gives you the desired term (e.g., the term with a specific power of $x$ or $y$).
• Calculate Binomial Coefficients: Remember how to calculate $\binom{n}{k}$. Practice with factorials to get comfortable with the calculations.
• Simplify Carefully: Pay close attention to exponents and coefficients. Double-check your calculations to avoid errors.
• Break it Down: If the problem seems complex, break it down into smaller, more manageable steps. Work methodically.
• Practice, Practice, Practice: The more problems you solve, the better you'll become at recognizing patterns and applying the binomial theorem.
• Check Your Work: Always double-check your answer to make sure it makes sense in the context of the problem.

By keeping these tips in mind and practicing regularly, you'll be well-equipped to tackle any binomial expansion problem that comes your way. Remember, it’s all about breaking down the problem, understanding the concepts, and being patient with yourself. Good luck, and keep practicing, guys!

Conclusion: Mastering the Binomial Theorem

So, there you have it! We've successfully found the coefficient of the $x^9y$ term in the binomial expansion of $\left(2y + 4x^3\right)^4$. We walked through the binomial theorem, carefully calculated the terms, and arrived at the correct answer: 512. This problem showcases the power of mathematical formulas and the importance of breaking down complex problems into manageable steps. Keep in mind, that mathematics is all about practice and understanding. The more you practice, the more confident you'll become. This problem shows us the importance of understanding the underlying mathematical principles and how to apply them systematically. Keep practicing, and you'll become a pro at binomial expansions in no time! Don’t forget to review the key concepts and formulas we used. Happy calculating!