Which Of The Following Is The Expansion Of $(2m-n)^7$?A. $128m^7 + 448m^6n + 672m^5n^2 + 560m^4n^3 + 280m^3n^4 + 84m^2n^5 + 14mn^6 + N^7$B. \$128m^7 + 14m^6n + 42m^5n^2 + 70m^4n^3 + 70m^3n^4 + 42m^2n^5 + 14mn^6 +

Introduction

Binomial expressions are a fundamental concept in algebra, and understanding how to expand them is crucial for solving various mathematical problems. In this article, we will explore the expansion of binomial expressions, focusing on the specific case of $(2m-n)^7$. We will examine the correct expansion of this expression and compare it with the given options.

The Binomial Theorem

The binomial theorem is a powerful tool for expanding binomial expressions. It states that for any positive integer $n$, the expansion of $(a+b)^n$ is given by:

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

where $\binom{n}{k}$ is the binomial coefficient, defined as:

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

Expanding $(2m-n)^7$

To expand $(2m-n)^7$, we can use the binomial theorem. We have:

$$(2m-n)^7 = (2m)^7 - 7(2m)^6n + 21(2m)^5n^2 - 35(2m)^4n^3 + 35(2m)^3n^4 - 21(2m)^2n^5 + 7(2m)n^6 - n^7$$

Using the binomial theorem, we can rewrite this expression as:

$$(2m-n)^7 = \sum_{k=0}^{7} \binom{7}{k} (2m)^{7-k} (-n)^k$$

Evaluating the binomial coefficients and simplifying the expression, we get:

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

Comparing with the Given Options

Now, let's compare the correct expansion of $(2m-n)^7$ with the given options:

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

B. $128m^7 + 14m^6n + 42m^5n^2 + 70m^4n^3 + 70m^3n^4 + 42m^2n^5 + 14mn^6 + n^7$

It is clear that option A is the correct expansion of $(2m-n)^7$. The coefficients and terms in option A match the correct expansion, while option B is incorrect.

Conclusion

In conclusion, the expansion of $(2m-n)^7$ is given by:

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

This expansion can be obtained using the binomial theorem, and it is essential to understand this concept to solve various mathematical problems. By comparing the correct expansion with the given options, we can see that option A is the correct answer.

Final Answer

The final answer is:

$$

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool for expanding binomial expressions. It states that for any positive integer $n$, the expansion of $(a+b)^n$ is given by:

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

where $\binom{n}{k}$ is the binomial coefficient, defined as:

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

Q: How do I apply the binomial theorem to expand a binomial expression?

A: To apply the binomial theorem, you need to:

1. Identify the binomial expression you want to expand.
2. Determine the value of $n$.
3. Use the formula $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ to expand the expression.
4. Simplify the resulting expression.

Q: What is the difference between the binomial theorem and the binomial expansion?

A: The binomial theorem and the binomial expansion are related but distinct concepts.

• The binomial theorem is a formula for expanding binomial expressions.
• The binomial expansion is the result of applying the binomial theorem to a specific binomial expression.

Q: Can I use the binomial theorem to expand expressions with negative coefficients?

A: Yes, you can use the binomial theorem to expand expressions with negative coefficients. However, you need to be careful when simplifying the resulting expression.

For example, consider the expression $(2m-n)^7$. To expand this expression, you can use the binomial theorem:

$$(2m-n)^7 = (2m)^7 - 7(2m)^6n + 21(2m)^5n^2 - 35(2m)^4n^3 + 35(2m)^3n^4 - 21(2m)^2n^5 + 7(2m)n^6 - n^7$$

Q: How do I determine the correct expansion of a binomial expression?

A: To determine the correct expansion of a binomial expression, you need to:

1. Apply the binomial theorem to the expression.
2. Simplify the resulting expression.
3. Compare the resulting expression with the given options.

Q: What are some common mistakes to avoid when expanding binomial expressions?

A: Some common mistakes to avoid when expanding binomial expressions include:

• Failing to apply the binomial theorem correctly.
• Simplifying the resulting expression incorrectly.
• Not comparing the resulting expression with the given options.

Q: Can I use the binomial theorem to expand expressions with fractional coefficients?

A: Yes, you can use the binomial theorem to expand expressions with fractional coefficients. However, you need to be careful when simplifying the resulting expression.

For example, consider the expression $(\frac{1}{2}m + \frac{1}{3}n)^7$. To expand this expression, you can use the binomial theorem:

$$(\frac{1}{2}m + \frac{1}{3}n)^7 = (\frac{1}{2}m)^7 + 7(\frac{1}{2}m)^6(\frac{1}{3}n) + 21(\frac{1}{2}m)^5(\frac{1}{3}n)^2 + ...$$

Conclusion

In conclusion, the binomial theorem is a powerful tool for expanding binomial expressions. By understanding how to apply the binomial theorem and avoiding common mistakes, you can determine the correct expansion of a binomial expression. Remember to simplify the resulting expression and compare it with the given options to ensure accuracy.

Final Tips

• Practice applying the binomial theorem to different binomial expressions.
• Pay attention to the signs and coefficients in the resulting expression.
• Use the binomial theorem to expand expressions with negative and fractional coefficients.

By following these tips and understanding the binomial theorem, you can become proficient in expanding binomial expressions and solving mathematical problems with confidence.