Write The Binomial Expansion Of $(x+2y)^5$.

Introduction

The binomial expansion is a fundamental concept in algebra and mathematics, used to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are constants or variables, and $n$ is a positive integer. In this article, we will focus on the binomial expansion of the expression $(x+2y)^5$. We will explore the concept of binomial expansion, its formula, and how to apply it to expand the given expression.

What is Binomial Expansion?

Binomial expansion is a mathematical technique used to expand expressions of the form $(a+b)^n$. It is a way of expressing a binomial raised to a power as a sum of terms, each term being a combination of the two original variables. The binomial expansion formula is given by:

$$(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$$

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

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

The Binomial Expansion Formula

The binomial expansion formula is a powerful tool for expanding expressions of the form $(a+b)^n$. It is based on the concept of combinations, and it allows us to express a binomial raised to a power as a sum of terms, each term being a combination of the two original variables.

To apply the binomial expansion formula, we need to follow these steps:

1. Identify the values of $a$, $b$, and $n$ in the given expression.
2. Calculate the binomial coefficients using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Substitute the values of $a$, $b$, and the binomial coefficients into the binomial expansion formula.
4. Simplify the resulting expression to obtain the expanded form.

Expanding $(x+2y)^5$

Now that we have a good understanding of the binomial expansion formula, let's apply it to expand the expression $(x+2y)^5$. We will follow the steps outlined above to obtain the expanded form.

Step 1: Identify the values of $a$, $b$, and $n$

In the expression $(x+2y)^5$, we have:

• $a = x$
• $b = 2y$
• $n = 5$

Step 2: Calculate the binomial coefficients

Using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, we can calculate the binomial coefficients as follows:

• $\binom{5}{0} = \frac{5!}{0!(5-0)!} = 1$
• $\binom{5}{1} = \frac{5!}{1!(5-1)!} = 5$
• $\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10$
• $\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10$
• $\binom{5}{4} = \frac{5!}{4!(5-4)!} = 5$
• $\binom{5}{5} = \frac{5!}{5!(5-5)!} = 1$

Step 3: Substitute the values of $a$, $b$, and the binomial coefficients into the binomial expansion formula

Substituting the values of $a$, $b$, and the binomial coefficients into the binomial expansion formula, we get:

$$(x+2y)^5 = \binom{5}{0} x^5 (2y)^0 + \binom{5}{1} x^4 (2y)^1 + \binom{5}{2} x^3 (2y)^2 + \binom{5}{3} x^2 (2y)^3 + \binom{5}{4} x^1 (2y)^4 + \binom{5}{5} x^0 (2y)^5$$

Step 4: Simplify the resulting expression

Simplifying the resulting expression, we get:

$$(x+2y)^5 = x^5 + 5x^4(2y) + 10x^3(2y)^2 + 10x^2(2y)^3 + 5x(2y)^4 + (2y)^5$$

Expanding the terms, we get:

$$(x+2y)^5 = x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5$$

Conclusion

In this article, we have explored the concept of binomial expansion, its formula, and how to apply it to expand the expression $(x+2y)^5$. We have followed the steps outlined above to obtain the expanded form, which is:

$$(x+2y)^5 = x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5$$

Q&A: Binomial Expansion

In this article, we will answer some of the most frequently asked questions about binomial expansion. Whether you are a student, a teacher, or simply someone who wants to learn more about binomial expansion, this article is for you.

Q: What is binomial expansion?

A: Binomial expansion is a mathematical technique used to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are constants or variables, and $n$ is a positive integer.

Q: What is the binomial expansion formula?

A: The binomial expansion formula is given by:

$$(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$$

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the binomial expansion formula. It is defined as:

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

Q: How do I apply the binomial expansion formula?

A: To apply the binomial expansion formula, you need to follow these steps:

1. Identify the values of $a$, $b$, and $n$ in the given expression.
2. Calculate the binomial coefficients using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Substitute the values of $a$, $b$, and the binomial coefficients into the binomial expansion formula.
4. Simplify the resulting expression to obtain the expanded form.

Q: What is the expanded form of $(x+2y)^5$?

A: The expanded form of $(x+2y)^5$ is:

$$(x+2y)^5 = x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5$$

Q: Can I use binomial expansion to expand expressions with negative exponents?

A: Yes, you can use binomial expansion to expand expressions with negative exponents. However, you need to be careful when dealing with negative exponents, as they can lead to complex expressions.

Q: Can I use binomial expansion to expand expressions with fractional exponents?

A: Yes, you can use binomial expansion to expand expressions with fractional exponents. However, you need to be careful when dealing with fractional exponents, as they can lead to complex expressions.

Q: What are some common applications of binomial expansion?

A: Binomial expansion has many common applications in mathematics, science, and engineering. Some of the most common applications include:

• Calculating probabilities and statistics
• Modeling population growth and decay
• Analyzing electrical circuits and networks
• Solving optimization problems

Conclusion

In this article, we have answered some of the most frequently asked questions about binomial expansion. Whether you are a student, a teacher, or simply someone who wants to learn more about binomial expansion, we hope that this article has provided you with a comprehensive guide to this important mathematical technique.

Additional Resources

If you want to learn more about binomial expansion, we recommend the following resources:

• Khan Academy: Binomial Expansion
• Mathway: Binomial Expansion
• Wolfram Alpha: Binomial Expansion

We hope that this article has been helpful in your understanding of binomial expansion. If you have any further questions or need additional help, please don't hesitate to ask.