8th Term Of (x+y)^10: A Step-by-Step Guide

Hey guys! Ever wondered how to expand those binomial expressions without manually multiplying everything out? Binomial expansions can seem intimidating, but they're actually super manageable once you grasp the underlying principles. In this article, we're going to break down the process of finding a specific term in a binomial expansion, using the example of finding the 8th term of $(x+y)^{10}$. Let's dive in and make this concept crystal clear!

Understanding Binomial Expansion

Before we jump into finding the 8th term, let's quickly recap what binomial expansion is all about. A binomial is simply an algebraic expression with two terms, like $(x+y)$. When we raise a binomial to a power, such as $(x+y)^{10}$, we're essentially multiplying the binomial by itself that many times. Expanding this manually would be incredibly tedious, especially for higher powers! That's where the binomial theorem comes to our rescue. The binomial theorem provides a formula to directly calculate the expansion without repeated multiplication.

The binomial theorem states that for any non-negative integer n:

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

Where $\binom{n}{k}$ is the binomial coefficient, also known as "n choose k", and it's calculated as:

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

Don't let the notation scare you! Let's break it down:

• n! represents the factorial of n, which is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
• k is the term number we're looking for (starting from 0).
• The binomial coefficient $\binom{n}{k}$ tells us the numerical coefficient of the term.

The Power of the Binomial Theorem

The binomial theorem is a powerful tool because it allows us to:

• Expand binomials raised to any power n without manual multiplication.
• Find a specific term in the expansion without calculating all the preceding terms. This is exactly what we'll be doing in this article!
• Understand the patterns and relationships within binomial expansions.

To truly appreciate the binomial theorem, consider the expansion of $(x + y)^3$:

$(x + y)^3 = \binom{3}{0}x^3y^0 + \binom{3}{1}x^2y^1 + \binom{3}{2}x^1y^2 + \binom{3}{3}x^0y^3$

Let's calculate the binomial coefficients:

• $\binom{3}{0} = \frac{3!}{0!3!} = 1$
• $\binom{3}{1} = \frac{3!}{1!2!} = 3$
• $\binom{3}{2} = \frac{3!}{2!1!} = 3$
• $\binom{3}{3} = \frac{3!}{3!0!} = 1$

Plugging these back into the expansion, we get:

$(x + y)^3 = 1x^3y^0 + 3x^2y^1 + 3x^1y^2 + 1x^0y^3 = x^3 + 3x^2y + 3xy^2 + y^3$

This demonstrates how the binomial coefficients determine the numerical coefficients in the expansion. The pattern of these coefficients can be visualized using Pascal's Triangle, which provides a geometric way to calculate binomial coefficients for different powers. Each number in Pascal's Triangle is the sum of the two numbers directly above it. The rows of the triangle correspond to the powers of the binomial, starting with power 0 at the top.

Connecting the Binomial Theorem to Our Problem

Now that we have a solid understanding of the binomial theorem, we can apply it to our specific problem: finding the 8th term of $(x+y)^{10}$. Remember, the formula from the binomial theorem is our key to unlocking this problem. By carefully identifying the values of n and k and applying the formula, we can efficiently find the desired term. In the next sections, we will go through the step-by-step process to calculate the 8th term, making sure you understand each step along the way. So, let's move on to the next section and see how this works in practice!

Finding the 8th Term: A Step-by-Step Approach

Okay, let's get to the core of the problem: determining the 8th term in the expansion of $(x+y)^{10}$. Remember that binomial theorem formula we talked about? That's our trusty tool here.

The general term in the binomial expansion of $(x+y)^n$ is given by:

$T_{k+1} = \binom{n}{k} x^{n-k} y^k$

Notice the k+1 subscript. This is crucial! It means that if we want the 8th term, we need to set k+1 equal to 8, which means k will be 7. It's a common mistake to use k = 8 directly, so always double-check this.

Step 1: Identify n and k

In our case, we have:

• n = 10 (the power to which the binomial is raised)
• k = 7 (since we're looking for the 8th term, and k+1 = 8)

Step 2: Plug the Values into the Formula

Now we substitute these values into the general term formula:

$T_{8} = T_{7+1} = \binom{10}{7} x^{10-7} y^7$

This simplifies to:

$T_{8} = \binom{10}{7} x^3 y^7$

See? We're already getting somewhere! The exponents of x and y are starting to reveal themselves. Now, the only thing left to do is calculate the binomial coefficient.

Step 3: Calculate the Binomial Coefficient

Remember the formula for the binomial coefficient:

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

Let's plug in our values:

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

Now, let's expand the factorials:

$\binom{10}{7} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)(3 \times 2 \times 1)}$

Notice how we can cancel out the 7! from both the numerator and the denominator:

$\binom{10}{7} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1}$

This simplifies the calculation significantly. Now we can do some further simplification:

$\binom{10}{7} = \frac{10 \times (3 \times 3) \times (4 \times 2)}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$

So, the binomial coefficient $\binom{10}{7}$ is 120.

Step 4: Put It All Together

Now we have all the pieces we need! Let's substitute the binomial coefficient back into our expression for the 8th term:

$T_{8} = 120 x^3 y^7$

And there you have it! The 8th term in the expansion of $(x+y)^{10}$ is $120x^3y^7$.

Reflecting on the Process

By breaking down the problem into manageable steps, we were able to efficiently find the 8th term. Remember, the key is to:

1. Identify n and k carefully.
2. Substitute the values into the general term formula.
3. Calculate the binomial coefficient (don't be afraid to simplify factorials!).
4. Combine all the pieces to get the final term.

This step-by-step approach will help you tackle similar problems with confidence. In the next section, we'll solidify your understanding with some practice questions and explore some common pitfalls to avoid.

Practice and Common Mistakes

Alright, guys, let's put your newfound knowledge to the test! Practice is key to mastering binomial expansions. We'll also look at some common mistakes people make so you can steer clear of them.

Practice Questions

Here are a couple of practice questions to try out:

1. What is the 6th term of the expansion $(a+b)^9$?
2. Find the 4th term of the expansion $(2x - 1)^7$.

Take your time, follow the steps we outlined earlier, and see if you can nail these. Remember to carefully identify n and k, calculate the binomial coefficient, and then combine everything to get the final term. Don't hesitate to go back and review the previous sections if you need a refresher.

Hint for Question 2: Remember that the binomial has a subtraction sign. This will affect the sign of the term in the expansion.

Common Mistakes to Avoid

Now, let's talk about some common pitfalls that students often encounter when dealing with binomial expansions. Being aware of these mistakes can save you a lot of trouble.

1. Incorrectly Identifying k: This is probably the most frequent error. Remember that the term number is k+1, not k. So, if you're looking for the 8th term, k is 7, not 8. Always double-check this before proceeding!
2. Forgetting the Factorials: The binomial coefficient formula involves factorials, and it's easy to make a mistake if you rush through the calculation. Make sure you expand the factorials correctly and cancel out common terms to simplify the process.
3. Sign Errors: When dealing with binomials that have a subtraction sign, like $(2x - 1)^7$, pay close attention to the signs. The terms will alternate in sign, so you need to be careful about whether the term you're looking for will be positive or negative.
4. Incorrectly Applying the Formula: Double-check that you're using the correct formula and that you're substituting the values in the right places. It's a good idea to write out the formula before you start plugging in numbers.
5. Not Simplifying: After calculating the binomial coefficient and substituting it into the term, make sure you simplify the expression as much as possible. This includes simplifying factorials, combining like terms, and reducing fractions.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence when working with binomial expansions. Now, let's look at the solutions to the practice questions.

Solutions to Practice Questions

Let's break down the solutions to the practice questions so you can check your work and see where you might have gone wrong.

Question 1: What is the 6th term of the expansion $(a+b)^9$?

• n = 9
• k = 5 (since we want the 6th term, k+1 = 6)

Using the general term formula:

$T_{6} = \binom{9}{5} a^{9-5} b^5 = \binom{9}{5} a^4 b^5$

Now, let's calculate the binomial coefficient:

$\binom{9}{5} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$

So, the 6th term is $126a^4b^5$.

Question 2: Find the 4th term of the expansion $(2x - 1)^7$.

• n = 7
• k = 3 (since we want the 4th term, k+1 = 4)

Using the general term formula:

$T_{4} = \binom{7}{3} (2x)^{7-3} (-1)^3 = \binom{7}{3} (2x)^4 (-1)^3$

Now, let's calculate the binomial coefficient:

$\binom{7}{3} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

Substitute back into the expression:

$T_{4} = 35 (2x)^4 (-1)^3 = 35 (16x^4) (-1) = -560x^4$

So, the 4th term is $-560x^4$. Notice the negative sign! This is crucial when dealing with binomials that have a subtraction sign.

Key Takeaways from Practice

By working through these practice questions, you've hopefully gained a better understanding of how to apply the binomial theorem to find specific terms in an expansion. Remember to be meticulous with your calculations, pay attention to signs, and always double-check your work. With practice, you'll become more comfortable and confident in your ability to tackle these types of problems. In our final section, we'll wrap up with a summary of the key points and discuss the real-world applications of binomial expansions.

Conclusion: Mastering Binomial Expansions and Their Applications

Well, guys, we've covered a lot in this article! We started with the basics of binomial expansion, learned about the binomial theorem, walked through a step-by-step process for finding specific terms, tackled some practice questions, and even discussed common mistakes to avoid. By now, you should have a solid understanding of how to find the 8th term (or any term!) in a binomial expansion.

Key Takeaways

Let's quickly recap the most important points:

• Binomial Theorem: This is the foundation of binomial expansions. Remember the formula: $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$.
• General Term Formula: The general term in the expansion is $T_{k+1} = \binom{n}{k} x^{n-k} y^k$. This is your go-to formula for finding a specific term.
• Identifying n and k: n is the power of the binomial, and k is one less than the term number you're looking for.
• Binomial Coefficient: Calculate $\binom{n}{k}$ using the formula $\frac{n!}{k!(n-k)!}$. Simplify factorials to make the calculation easier.
• Common Mistakes: Watch out for incorrect k values, factorial errors, sign errors, and not simplifying the final expression.

By keeping these key takeaways in mind, you'll be well-equipped to handle binomial expansion problems with confidence.

Real-World Applications of Binomial Expansions

You might be wondering, "Okay, this is cool, but where does this stuff actually get used?" Binomial expansions aren't just abstract mathematical concepts; they have practical applications in various fields.

1. Probability and Statistics: Binomial expansions are fundamental in probability theory. They're used to calculate probabilities in situations where there are two possible outcomes, such as coin flips or success/failure scenarios. For example, the binomial distribution, which describes the probability of obtaining a certain number of successes in a fixed number of trials, is directly derived from the binomial theorem.
2. Computer Science: In computer science, binomial expansions are used in algorithms for data compression and error correction. They also play a role in the analysis of algorithms and the design of efficient data structures.
3. Finance: Binomial models are used in finance to price options and other financial derivatives. These models use binomial trees to represent the possible paths of an asset price over time, and the binomial theorem helps in calculating the probabilities associated with each path.
4. Physics and Engineering: Binomial expansions are used in physics and engineering to approximate complex expressions. For example, the binomial theorem can be used to approximate $(1+x)^n$ when x is small, which is useful in various physical calculations.
5. Calculus: Binomial expansions are used in calculus to derive power series representations of functions. Power series are infinite sums that can be used to approximate the values of functions, and the binomial theorem provides a way to find these series for certain types of functions.

These are just a few examples of the many ways in which binomial expansions are used in the real world. As you continue your studies in mathematics and related fields, you'll likely encounter even more applications of this powerful tool.

Final Thoughts

Mastering binomial expansions is a valuable skill that will serve you well in various areas of mathematics and beyond. By understanding the binomial theorem, practicing the steps involved in finding specific terms, and being aware of common mistakes, you can confidently tackle these problems. So, keep practicing, keep exploring, and keep expanding your mathematical horizons!

I hope this article has been helpful and has made binomial expansions a little less intimidating. Keep up the great work, and I'll see you in the next math adventure!