Terms In The Expansion Of (x+y)^10 A Mathematical Exploration

Hey guys! Let's dive into an exciting mathematical problem today. We're going to explore the expansion of the binomial expression $(x+y)^{10}$. Our main goal is to figure out which terms will actually appear when we expand this expression. Each term will have a real constant, which we'll call 'a'. The options we need to consider are: A. $ax^5y^5$, B. $ax^3y^7$, C. $ax^5$, and D. $ay^5$. So, let's put on our mathematical hats and get started!

Understanding the Binomial Theorem

Before we jump into the specifics, it's super important to have a solid understanding of the binomial theorem. The binomial theorem is our trusty tool for expanding expressions like $(x+y)^n$, where 'n' is a non-negative integer. This theorem gives us a systematic way to determine the coefficients and exponents in the expansion. Think of it as a roadmap that guides us through the expansion process, ensuring we don't get lost in the algebraic wilderness. The general formula for the binomial theorem is:

$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

Where $\binom{n}{k}$ represents the binomial coefficient, often read as "n choose k," and it's calculated as:

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

Here, '!' denotes the factorial, which means multiplying a number by all the positive integers less than it (e.g., 5! = 5 × 4 × 3 × 2 × 1). Now, let's break down this formula. The sigma notation ($\sum$) tells us we're summing terms. The index 'k' starts at 0 and goes all the way up to 'n'. For each value of 'k', we calculate a term in the expansion. This term consists of three parts: the binomial coefficient $\binom{n}{k}$, the x term raised to the power of $n-k$ ($x^{n-k}$), and the y term raised to the power of 'k' ($y^k$). The binomial coefficient $\binom{n}{k}$ gives us the numerical coefficient for each term. It tells us how many ways we can choose 'k' items from a set of 'n' items. The exponents of 'x' and 'y' change with each term, but their sum always equals 'n'. This is a crucial observation that will help us solve our problem. So, in a nutshell, the binomial theorem provides a structured way to expand binomial expressions, giving us both the coefficients and the variable terms. It's a powerful tool that simplifies what could otherwise be a tedious and error-prone process. Remember, the key is to understand the formula and how each part contributes to the overall expansion. Once you've grasped this, you're well-equipped to tackle a wide range of binomial expansion problems.

Applying the Binomial Theorem to (x+y)^10

Okay, now that we've got a handle on the binomial theorem, let's apply it to our specific problem: expanding $(x+y)^{10}$. In this case, 'n' is 10. This means we're going to have terms where the exponents of 'x' and 'y' add up to 10. Remember that key takeaway from our discussion about the binomial theorem? The sum of the exponents must equal 'n'. So, let's think about what possible combinations of exponents for 'x' and 'y' could show up in the expansion. We'll have terms like $x^{10}y^0$, $x^9y^1$, $x^8y^2$, and so on, all the way up to $x^0y^{10}$. Each of these terms will also have a binomial coefficient in front of it, which we get from the $\binom{10}{k}$ part of the binomial theorem formula. Now, let's consider each of the options given in the question and see if they fit this pattern. We'll check if the exponents add up to 10. If they do, then that term is a possible part of the expansion. If they don't, then we can rule it out right away. This is a straightforward way to narrow down our choices. We're essentially using the binomial theorem as a filter, allowing only those terms that meet the criteria to pass through. By systematically checking each option, we can confidently determine which terms will appear in the expansion of $(x+y)^{10}$. So, let's get started with analyzing the options and see which ones make the cut!

Analyzing the Options

Alright, let's get down to business and analyze each of the options to see which terms appear in the expansion of $(x+y)^{10}$. Remember, our golden rule is that the exponents of 'x' and 'y' must add up to 10.

A. $ax^5y^5$

First up, we have $ax^5y^5$. Let's check those exponents. We have 5 for 'x' and 5 for 'y'. Adding them together, 5 + 5 = 10. Bingo! This term satisfies our rule. So, $ax^5y^5$ is indeed a term that will appear in the expansion of $(x+y)^{10}$. To be absolutely sure, we can also calculate the binomial coefficient for this term. Here, the exponent of 'y' is 5, so k = 5. The binomial coefficient is $\binom{10}{5} = \frac{10!}{5!5!} = 252$. This tells us that the term will actually be $252x^5y^5$. The 'a' in our option simply represents this real constant (252 in this case).

B. $ax^3y^7$

Next, we have $ax^3y^7$. Let's do the exponent check again. We have 3 for 'x' and 7 for 'y'. Adding them together, 3 + 7 = 10. Another winner! This term also satisfies our rule. Thus, $ax^3y^7$ is a term that will appear in the expansion of $(x+y)^{10}$. Just like before, we can find the specific coefficient using the binomial theorem. Here, k = 7 (the exponent of 'y'), so the binomial coefficient is $\binom{10}{7} = \frac{10!}{7!3!} = 120$. This means the term is $120x^3y^7$, and 'a' represents 120 in this case.

C. $ax^5$

Now, let's look at $ax^5$. This one's a bit different because it only shows 'x' and doesn't explicitly show a 'y' term. But we can think of it as $ax^5y^0$ or, better yet, realize that for the exponents to add up to 10, there should be a $y^5$ term. So, the exponent of 'y' is implicitly 0. Adding the exponents, 5 + 0 = 5. Oops! This does not equal 10. Therefore, $ax^5$ is not a term that will appear in the expansion of $(x+y)^{10}$. We can rule this one out right away.

D. $ay^5$

Finally, we have $ay^5$. Similar to option C, this term only explicitly shows 'y'. We can think of it as $ax^0y^5$. To make the exponents add up to 10, we would need $x^{10}$, however this option doesn't include the $x^5$ term. Adding the exponents as they implicitly are, 0 + 5 = 5. This also does not equal 10. So, $ay^5$ is not a term that will appear in the expansion of $(x+y)^{10}$. We can eliminate this option as well.

Conclusion - Which Terms Appear?

Alright guys, we've done the work, crunched the numbers, and analyzed each option. Let's bring it all together and see what we've found. We started with the expression $(x+y)^{10}$ and wanted to know which of the given terms would appear in its expansion. We used the binomial theorem as our guide, focusing on the crucial fact that the exponents of 'x' and 'y' in each term must add up to 10.

We methodically checked each option:

• Option A, $ax^5y^5$: The exponents 5 and 5 added up to 10. This one does appear in the expansion.
• Option B, $ax^3y^7$: The exponents 3 and 7 added up to 10. This one also appears in the expansion.
• Option C, $ax^5$: The exponents 5 and (implicitly) 0 added up to 5, not 10. This one does not appear.
• Option D, $ay^5$: The exponents (implicitly) 0 and 5 added up to 5, not 10. This one does not appear either.

So, the final answer is that the terms $ax^5y^5$ and $ax^3y^7$ will appear in the expansion of $(x+y)^{10}$. We successfully used the binomial theorem to navigate this problem and arrive at the correct solution. Great job, everyone! This exercise highlights the power of understanding fundamental mathematical principles and applying them systematically to solve problems. Keep practicing, and you'll become a pro at these types of expansions in no time! Remember, math can be challenging, but it's also incredibly rewarding when you unlock a solution. And the best part is, the more you learn, the more you realize how interconnected everything is in the world of mathematics. So, keep exploring, keep questioning, and keep learning!