Expanding (2m-n)^7: A Mathematical Exploration

Alright, guys, let's dive into the fascinating world of binomial expansions! Specifically, we're going to figure out how to expand $(2m-n)^7$. This might seem daunting at first, but with a little bit of binomial theorem magic, we can break it down step by step. So grab your calculators (or your mental math muscles) and let's get started!

Understanding the Binomial Theorem

Before we jump into the expansion of $(2m-n)^7)$, let's quickly refresh our understanding of the binomial theorem. The binomial theorem provides a formula for expanding expressions of the form $(a+b)^n$, where n is a non-negative integer. The theorem states that:

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

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

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

Here, n! denotes the factorial of n, which is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

The binomial theorem essentially tells us how to find each term in the expansion. Each term consists of a binomial coefficient, a power of the first term (a), and a power of the second term (b). The binomial coefficients determine the numerical factors in each term, and the powers of a and b determine the variables and their exponents. Understanding this theorem is crucial for expanding expressions like $(2m-n)^7$ efficiently and accurately. It provides a systematic way to account for all the terms and their corresponding coefficients.

Applying the Binomial Theorem to (2m-n)^7

Now that we've got the binomial theorem fresh in our minds, let's apply it to our specific expression: $(2m-n)^7)$. Here, a = 2m, b = -n, and n = 7. We're going to calculate each term in the expansion using the binomial theorem formula. Remember, each term will have the form ${7 \choose k} (2m)^{7-k} (-n)^k$, where k ranges from 0 to 7.

Let's break it down term by term:

• k = 0: ${7 \choose 0} (2m)^7 (-n)^0 = 1 * (128m^7) * 1 = 128m^7$
• k = 1: ${7 \choose 1} (2m)^6 (-n)^1 = 7 * (64m^6) * (-n) = -448m^6n$
• k = 2: ${7 \choose 2} (2m)^5 (-n)^2 = 21 * (32m^5) * (n^2) = 672m^5n^2$
• k = 3: ${7 \choose 3} (2m)^4 (-n)^3 = 35 * (16m^4) * (-n^3) = -560m^4n^3$
• k = 4: ${7 \choose 4} (2m)^3 (-n)^4 = 35 * (8m^3) * (n^4) = 280m^3n^4$
• k = 5: ${7 \choose 5} (2m)^2 (-n)^5 = 21 * (4m^2) * (-n^5) = -84m^2n^5$
• k = 6: ${7 \choose 6} (2m)^1 (-n)^6 = 7 * (2m) * (n^6) = 14mn^6$
• k = 7: ${7 \choose 7} (2m)^0 (-n)^7 = 1 * (1) * (-n^7) = -n^7$

Notice the alternating signs due to the (-n) term. Make sure to keep track of those negative signs, guys, as they are crucial for getting the correct expansion!

Combining the Terms

Now that we have calculated each term individually, we need to combine them to get the full expansion of $(2m-n)^7)$. This involves simply adding all the terms we calculated in the previous step. So, let's put it all together:

$(2m-n)^7 = 128m^7 - 448m^6n + 672m^5n^2 - 560m^4n^3 + 280m^3n^4 - 84m^2n^5 + 14mn^6 - n^7$

This is the expanded form of $(2m-n)^7)$. As you can see, it's a polynomial with terms involving different powers of m and n, along with their corresponding coefficients. This expansion can be used in various mathematical contexts, such as simplifying expressions, solving equations, or approximating functions. Understanding the binomial theorem and its application allows us to efficiently expand expressions like this without having to manually multiply out the factors. It's a powerful tool in algebra and calculus, so mastering it is definitely worth the effort!

Common Mistakes to Avoid

When expanding binomials, it's easy to make a few common mistakes. Here's what to watch out for:

• Forgetting the Negative Sign: When b is negative (like in our case with -n), make sure you include the negative sign when calculating the terms. A missed negative sign can throw off the entire expansion.
• Incorrect Binomial Coefficients: Double-check your binomial coefficient calculations. A small error in calculating ${n \choose k}$ can lead to incorrect coefficients in your expansion.
• Incorrect Exponents: Make sure you correctly apply the exponents to both the numerical coefficients and the variables. For example, $(2m)^3$ is $8m^3$, not $2m^3$.
• Missing Terms: Ensure you calculate all the terms from k=0 to k=n. It's easy to accidentally skip a term, especially when dealing with larger exponents.

By being mindful of these potential pitfalls, you can increase your accuracy and avoid common errors when expanding binomials. Always double-check your work and pay attention to details!

Conclusion

Expanding $(2m-n)^7$ might seem like a beast at first, but with the power of the binomial theorem, it becomes a manageable task. Remember to break it down step by step, calculate each term carefully, and watch out for those sneaky negative signs! With practice, you'll become a binomial expansion pro in no time. Keep up the great work, guys, and happy expanding!

So the correct answer from the options would be none of the above, as the expansion is:

$128 m^7 - 448 m^6 n + 672 m^5 n^2 - 560 m^4 n^3 + 280 m^3 n^4 - 84 m^2 n^5 + 14 m n^6 - n^7$