Which Of The Following Terms Appear In The Expansion Of $(x+y)^{10}$? The Letter $a$ In Each Term Represents A Real Constant.A. $a X^3 Y^7$B. $a X^6 Y^4$C. $a X^4 Y^6$D. $a X^5$

Introduction

The binomial theorem is a fundamental concept in mathematics that provides a way to expand expressions of the form $(x+y)^n$, where $n$ is a positive integer. This theorem has numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will explore the binomial theorem and its applications, with a focus on the expansion of $(x+y)^{10}$.

The Binomial Theorem

The binomial theorem states that for any positive integer $n$, the expansion of $(x+y)^n$ is given by:

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

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

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

Expansion of $(x+y)^{10}$

Using the binomial theorem, we can expand $(x+y)^{10}$ as follows:

$$(x+y)^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} y^k$$

Evaluating the binomial coefficients, we get:

$$(x+y)^{10} = x^{10} + 10x^9y + 45x^8y^2 + 120x^7y^3 + 210x^6y^4 + 252x^5y^5 + 210x^4y^6 + 120x^3y^7 + 45x^2y^8 + 10xy^9 + y^{10}$$

Analyzing the Terms

Now, let's analyze the terms in the expansion of $(x+y)^{10}$:

• $x^{10}$: This term has a power of 10 for $x$ and a power of 0 for $y$.
• $10x^9y$: This term has a power of 9 for $x$ and a power of 1 for $y$.
• $45x^8y^2$: This term has a power of 8 for $x$ and a power of 2 for $y$.
• $120x^7y^3$: This term has a power of 7 for $x$ and a power of 3 for $y$.
• $210x^6y^4$: This term has a power of 6 for $x$ and a power of 4 for $y$.
• $252x^5y^5$: This term has a power of 5 for $x$ and a power of 5 for $y$.
• $210x^4y^6$: This term has a power of 4 for $x$ and a power of 6 for $y$.
• $120x^3y^7$: This term has a power of 3 for $x$ and a power of 7 for $y$.
• $45x^2y^8$: This term has a power of 2 for $x$ and a power of 8 for $y$.
• $10xy^9$: This term has a power of 1 for $x$ and a power of 9 for $y$.
• $y^{10}$: This term has a power of 0 for $x$ and a power of 10 for $y$.

Which Terms Appear in the Expansion?

Based on the analysis above, we can see that the following terms appear in the expansion of $(x+y)^{10}$:

• $a x^3 y^7$
• $a x^4 y^6$
• $a x^6 y^4$

These terms have powers of $x$ and $y$ that are consistent with the binomial theorem.

Conclusion

In conclusion, the binomial theorem provides a powerful tool for expanding expressions of the form $(x+y)^n$. By analyzing the terms in the expansion of $(x+y)^{10}$, we can see that the terms $a x^3 y^7$, $a x^4 y^6$, and $a x^6 y^4$ appear in the expansion. These terms have powers of $x$ and $y$ that are consistent with the binomial theorem.

Answer

The correct answer is:

• C. $a x^4 y^6$
• A. $a x^3 y^7$
• B. $a x^6 y^4$

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Q: What is the binomial theorem?

A: The binomial theorem is a fundamental concept in mathematics that provides a way to expand expressions of the form $(x+y)^n$, where $n$ is a positive integer.

Q: How is the binomial theorem used?

A: The binomial theorem is used in a wide range of applications, including algebra, geometry, and calculus. It is used to expand expressions, solve equations, and analyze functions.

Q: What is the formula for the binomial theorem?

A: The formula for the binomial theorem is:

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

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

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

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

A: To apply the binomial theorem to a problem, follow these steps:

1. Identify the expression you want to expand.
2. Determine the value of $n$.
3. Use the formula for the binomial theorem to expand the expression.
4. Simplify the resulting expression.

Q: What are some common mistakes to avoid when using the binomial theorem?

A: Some common mistakes to avoid when using the binomial theorem include:

• Forgetting to include the binomial coefficient.
• Making errors in the calculation of the binomial coefficient.
• Failing to simplify the resulting expression.

Q: Can the binomial theorem be used to expand expressions with negative exponents?

A: Yes, the binomial theorem can be used to expand expressions with negative exponents. However, you must be careful to handle the negative exponents correctly.

Q: How is the binomial theorem used in real-world applications?

A: The binomial theorem is used in a wide range of real-world applications, including:

• Finance: The binomial theorem is used to calculate the value of options and other financial instruments.
• Engineering: The binomial theorem is used to analyze the behavior of complex systems.
• Science: The binomial theorem is used to model the behavior of physical systems.

Q: Can the binomial theorem be used to solve equations?

A: Yes, the binomial theorem can be used to solve equations. However, you must be careful to handle the resulting expression correctly.

Q: How do I determine the value of the binomial coefficient?

A: The value of the binomial coefficient can be determined using the formula:

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

Q: Can the binomial theorem be used to analyze functions?

A: Yes, the binomial theorem can be used to analyze functions. However, you must be careful to handle the resulting expression correctly.

Q: How do I simplify the resulting expression after using the binomial theorem?

A: To simplify the resulting expression after using the binomial theorem, follow these steps:

1. Combine like terms.
2. Simplify the resulting expression.

Conclusion

In conclusion, the binomial theorem is a powerful tool for expanding expressions and solving equations. By understanding how to apply the binomial theorem and avoiding common mistakes, you can use this theorem to solve a wide range of problems in mathematics and real-world applications.