Fifth Term: (3x - 3y)^7 Expansion Explained

Hey guys! Ever found yourself staring at a binomial expansion and feeling like you're decoding ancient hieroglyphs? Fear not! Today, we're going to crack the code for finding a specific term in a binomial expansion – in this case, the fifth term in the expansion of $(3x - 3y)^7$. Buckle up, because we're about to dive into the fascinating world of binomial theorem!

Understanding the Binomial Theorem

Before we jump into the specifics of our problem, let's take a step back and understand the binomial theorem itself. This theorem provides a formula for expanding expressions of the form $(a + b)^n$, where 'n' is a non-negative integer. Think of it as a shortcut to avoid multiplying out $(a + b)$ by itself 'n' times – a process that can quickly become tedious and error-prone, especially when 'n' is a large number. The binomial theorem is your best friend when dealing with such expansions, offering a systematic way to determine each term in the expanded form. Understanding the core concepts of the binomial theorem is crucial for accurately finding specific terms within an expansion.

The binomial theorem states that:

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

Where:

• $(a + b)^n$ is the binomial expression raised to the power of 'n'.
• $\sum_{k=0}^{n}$ represents the summation from k = 0 to k = n, meaning we'll be adding up several terms.
• $\binom{n}{k}$ is the binomial coefficient, also known as "n choose k," which represents the number of ways to choose 'k' items from a set of 'n' items. It's calculated as $\frac{n!}{k!(n-k)!}$, where '!' denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1).
• $a^{n-k}$ is the first term ('a') raised to the power of (n - k).
• $b^k$ is the second term ('b') raised to the power of 'k'.

In essence, the theorem tells us that the expansion of $(a + b)^n$ will have (n + 1) terms, each with a specific coefficient and powers of 'a' and 'b'. The binomial coefficients, calculated using combinations, determine the numerical factor for each term. The exponents of 'a' decrease from 'n' to 0, while the exponents of 'b' increase from 0 to 'n'. This systematic pattern is what makes the binomial theorem so powerful and predictable.

Now, you might be wondering, "Why does this work?" The binomial theorem has its roots in combinatorics, the study of counting. Each term in the expansion corresponds to a specific combination of choosing 'b' from 'n' factors of $(a + b)`. When you multiply out the binomial expression, you're essentially distributing each term of one factor across all terms of the other factors. The binomial coefficients arise from counting the number of ways to obtain a particular combination of 'a' and 'b' terms. This combinatorial interpretation provides a deeper understanding of the theorem and its applicability.

Identifying the Fifth Term

Alright, now that we've got a handle on the binomial theorem, let's zero in on our main quest: finding the fifth term in the expansion of $(3x - 3y)^7$. The first crucial step is to correctly identify what 'n' and 'k' represent in this context. Remember, the general term in the binomial expansion is given by:

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

In our specific problem, we have:

• $n = 7$ (the exponent of the binomial)
• $a = 3x$ (the first term inside the binomial)
• $b = -3y$ (the second term inside the binomial – don't forget the negative sign!)

Now, here's a little trick to determine the value of 'k'. The first term in the expansion corresponds to k = 0, the second term to k = 1, the third term to k = 2, and so on. So, the fifth term will correspond to k = 4 (one less than the term number). This is a common point of confusion, so it's essential to remember this relationship between the term number and the value of 'k'. If you're ever unsure, you can always list out the first few terms and their corresponding 'k' values to confirm.

Why is it important to get the correct value of 'k'? Because 'k' dictates the powers of 'a' and 'b', as well as the binomial coefficient. A mistake in 'k' will lead to an incorrect term. For example, if we mistakenly used k = 5 instead of k = 4, we would be calculating the sixth term instead of the fifth. The binomial coefficients are also sensitive to 'k'; changing 'k' changes the number of combinations and, therefore, the coefficient.

So, with n = 7 and k = 4, we can now plug these values into the binomial theorem formula. We'll need to calculate the binomial coefficient $\binom{7}{4}$, as well as the powers of $(3x)$ and $(-3y)$. This process of identifying the correct 'n' and 'k' is the foundation for successfully finding any specific term in a binomial expansion. Once you've mastered this step, the rest is just arithmetic and careful application of the formula.

Calculating the Fifth Term

Okay, we've identified all the pieces of the puzzle. Now it's time to put them together and calculate the fifth term in the expansion of $(3x - 3y)^7$. Remember, the general term we're interested in is:

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

With our values: n = 7, k = 4, a = 3x, and b = -3y, we can substitute these into the formula:

Fifth Term = $\binom{7}{4} (3x)^{7-4} (-3y)^4$

Let's break this down step by step. First, we need to calculate the binomial coefficient $\binom{7}{4}$. Using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, we have:

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

So, the binomial coefficient $\binom{7}{4}$ is 35. This number represents the weight of this particular term in the expansion, reflecting the number of ways to choose 4 'b' terms from the 7 factors of (3x - 3y).

Next, let's deal with the powers of (3x) and (-3y). We have:

$(3x)^{7-4} = (3x)^3 = 3^3 x^3 = 27x^3$

and

$(-3y)^4 = (-3)^4 y^4 = 81y^4$

Notice how the exponent affects both the coefficient and the variable. When raising a product to a power, you raise each factor to that power. Also, remember that a negative number raised to an even power becomes positive, which is why $(-3)^4$ is positive 81. Keeping track of these details is crucial for accurate calculations.

Now, we can substitute these values back into our expression for the fifth term:

Fifth Term = $35 \times 27x^3 \times 81y^4$

Finally, we multiply the coefficients together:

Fifth Term = $35 \times 27 \times 81 x^3 y^4 = 76545 x^3 y^4$

Therefore, the fifth term in the expansion of $(3x - 3y)^7$ is $76545x^3y^4$. We've successfully navigated the binomial theorem, identified the correct term, and calculated its coefficient and variable parts. This process may seem lengthy at first, but with practice, you'll become much faster and more confident in your ability to tackle these types of problems.

Common Pitfalls and How to Avoid Them

Alright, guys, let's talk about some common traps that people fall into when working with the binomial theorem. Knowing these pitfalls will help you avoid making mistakes and ensure you get the correct answer every time. It is important to understand the concept of binomials so that you can avoid these common pitfalls.

1. Forgetting the Negative Sign: This is a big one! When your binomial has a subtraction, like in our example with $(3x - 3y)^7$, it's crucial to remember that the second term, 'b', includes the negative sign. So, in our case, b = -3y, not just 3y. Forgetting the negative sign will throw off your calculations, especially when raising 'b' to an even power. To avoid this, always explicitly write out the value of 'b' with its sign before plugging it into the formula. Double-check that you've accounted for the negative sign in your calculations.
2. Incorrect 'k' Value: As we discussed earlier, the term number and the value of 'k' are related but not the same. The 'k' value is always one less than the term number. So, for the fifth term, k = 4. A common mistake is to use k = 5 for the fifth term. To avoid this, always subtract 1 from the term number to get the correct 'k' value. If you're unsure, write out the term numbers and their corresponding 'k' values for the first few terms to solidify the pattern.
3. Miscalculating the Binomial Coefficient: The binomial coefficient can look intimidating with all those factorials, but it's just a matter of careful calculation. A common mistake is to incorrectly cancel out terms in the factorial expression. To avoid this, write out the factorials explicitly and then carefully cancel out common factors. It can also be helpful to use a calculator with a built-in combination function (usually denoted as nCr or similar) to double-check your calculations. Remember, the binomial coefficient represents the number of ways to choose 'k' items from 'n', so it should always be a whole number.
4. Incorrectly Applying Exponents: When raising a term like (3x) or (-3y) to a power, remember to apply the exponent to both the coefficient and the variable. For example, $(3x)^3 = 3^3 x^3 = 27x^3$, not just $3x^3$. To avoid this, break down the term into its factors and apply the exponent to each factor separately. Pay close attention to negative signs as well, as we discussed earlier.
5. Arithmetic Errors: Let's face it, even with a perfect understanding of the binomial theorem, arithmetic errors can still creep in. Multiplying large numbers and keeping track of exponents can be tricky. To minimize these errors, double-check your calculations at each step. Use a calculator to verify your arithmetic, especially for the final multiplication of the coefficients.

By being aware of these common pitfalls and taking steps to avoid them, you'll significantly improve your accuracy and confidence when working with the binomial theorem. Remember, practice makes perfect! The more you work through these types of problems, the more comfortable you'll become with the concepts and the calculations.

Practice Makes Perfect

So, there you have it! We've successfully navigated the binomial theorem to find the fifth term in the expansion of $(3x - 3y)^7$. We've covered the fundamentals of the theorem, identified the key values for our problem, performed the calculations, and even discussed common pitfalls to avoid. However, like any mathematical skill, mastery comes with practice. The more you work with the binomial theorem, the more comfortable and confident you'll become in applying it.

To truly solidify your understanding, I highly recommend working through additional examples. Try finding different terms in the expansion of various binomial expressions. Experiment with different values of 'n' and different terms inside the binomial. The more variety you encounter, the better you'll understand the nuances of the theorem and its applications. There are countless resources available online and in textbooks that provide practice problems and solutions. Don't hesitate to explore these resources and challenge yourself.

Consider these practice exercises:

• Find the third term in the expansion of $(2a + b)^6$.
• Determine the fourth term in the expansion of $(x - 2y)^5$.
• What is the coefficient of the term containing $x^2$ in the expansion of $(1 + x)^{10}$?

Working through these problems will not only reinforce your understanding of the binomial theorem but also help you develop problem-solving skills that are valuable in many areas of mathematics. Pay attention to the details, double-check your calculations, and don't be afraid to make mistakes – mistakes are learning opportunities! By identifying your errors and understanding why you made them, you'll be able to improve your accuracy and efficiency.

In addition to practice problems, try explaining the binomial theorem to someone else. Teaching is one of the best ways to learn. When you explain a concept to another person, you're forced to think about it in a clear and organized way. This process can reveal gaps in your own understanding and help you to solidify your knowledge. You can explain the theorem to a friend, a family member, or even an imaginary audience! The act of verbalizing the concepts and steps involved will make them more concrete in your mind.

So, grab a pencil, a piece of paper, and a calculator, and dive into the world of binomial expansions! With consistent practice and a solid understanding of the concepts, you'll be able to conquer any binomial theorem problem that comes your way. Keep up the great work, guys, and happy expanding!