The Binomial Expansion Of $(2x - Y)^3$ Is:A. $x^3 - 6x^2y + 12xy^2 - 8y^3$B. \$8x^3 - 12x^2y + 6xy^2 - Y^3$[/tex\]C. $2^3(x^3 - 3x^2y + 3xy^2 - Y^3)$D. $x^3 - 3x^2y + 3xy^2 - Y^3$

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Understanding the Binomial Expansion Formula

The binomial expansion is a mathematical formula used to expand expressions of the form $(a + b)^n$, where $a$ and $b$ are numbers or variables, and $n$ is a positive integer. The formula is given by:

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

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

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

Applying the Binomial Expansion Formula to $(2x - y)^3$

To expand the expression $(2x - y)^3$, we can use the binomial expansion formula with $a = 2x$, $b = -y$, and $n = 3$. Plugging these values into the formula, we get:

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

Calculating the Binomial Coefficients

To calculate the binomial coefficients, we can use the formula:

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

Plugging in the values $n = 3$ and $k = 0, 1, 2, 3$, we get:

$$\binom{3}{0} = \frac{3!}{0!(3-0)!} = 1$$

$$\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3$$

$$\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3$$

$$\binom{3}{3} = \frac{3!}{3!(3-3)!} = 1$$

Expanding the Expression

Now that we have calculated the binomial coefficients, we can expand the expression:

$$(2x - y)^3 = 1(2x)^3 (-y)^0 + 3(2x)^2 (-y)^1 + 3(2x)^1 (-y)^2 + 1(2x)^0 (-y)^3$$

$$= 1(8x^3) + 3(4x^2)(-y) + 3(2x)(-y)^2 + 1(-y)^3$$

$$= 8x^3 - 12x^2y + 6xy^2 - y^3$$

Conclusion

The binomial expansion of $(2x - y)^3$ is:

$$8x^3 - 12x^2y + 6xy^2 - y^3$$

This is option A in the given multiple-choice question.

Comparison with Other Options

Let's compare the correct expansion with the other options:

• Option B: $8x^3 - 12x^2y + 6xy^2 - y^3$ is incorrect because it has the wrong sign for the $y^3$ term.
• Option C: $2^3(x^3 - 3x^2y + 3xy^2 - y^3)$ is incorrect because it has the wrong coefficient for the $x^3$ term.
• Option D: $x^3 - 3x^2y + 3xy^2 - y^3$ is incorrect because it has the wrong coefficient for the $x^3$ term and the wrong sign for the $y^3$ term.

Final Answer

The binomial expansion of $(2x - y)^3$ is:

$$8x^3 - 12x^2y + 6xy^2 - y^3$$

This is option A in the given multiple-choice question.

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Introduction

In our previous article, we explored the binomial expansion of $(2x - y)^3$ and found that the correct expansion is:

$$8x^3 - 12x^2y + 6xy^2 - y^3$$

However, we know that there are many students who struggle with binomial expansions, and we want to help. In this article, we will answer some frequently asked questions about binomial expansions and provide additional examples to help you understand this important mathematical concept.

Q: What is a binomial expansion?

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

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 expression.
2. Calculate the binomial coefficients using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Plug the binomial coefficients into the formula $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$.
4. Simplify the expression to get the final answer.

Q: What are some common mistakes to avoid when applying the binomial expansion formula?

A: Here are some common mistakes to avoid:

• Not calculating the binomial coefficients correctly.
• Not plugging the binomial coefficients into the formula correctly.
• Not simplifying the expression correctly.
• Not checking the final answer for errors.

Q: Can you provide more examples of binomial expansions?

A: Here are some more examples of binomial expansions:

• $(x + y)^2 = x^2 + 2xy + y^2$
• $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$
• $(2x + 3y)^4 = 2^4x^4 + 4 \cdot 2^3x^3(3y) + 6 \cdot 2^2x^2(3y)^2 + 4 \cdot 2x(3y)^3 + (3y)^4$

Q: How do I use binomial expansions in real-life situations?

A: Binomial expansions have many real-life applications, including:

• Finance: Binomial expansions are used to calculate the value of options and other financial instruments.
• Engineering: Binomial expansions are used to calculate the stress and strain on materials.
• Science: Binomial expansions are used to calculate the probability of certain events occurring.

Conclusion

In this article, we have answered some frequently asked questions about binomial expansions and provided additional examples to help you understand this important mathematical concept. We hope that this article has been helpful in clarifying any confusion you may have had about binomial expansions.

Final Tips

• Practice, practice, practice: The more you practice binomial expansions, the more comfortable you will become with the formula and the more confident you will be in your ability to apply it.
• Use online resources: There are many online resources available that can help you learn binomial expansions, including video tutorials, practice problems, and interactive quizzes.
• Seek help when needed: If you are struggling with binomial expansions, don't be afraid to seek help from a teacher, tutor, or classmate.