What Is The Value Of The 3rd Term Of The Expansion $(3y - 2)^7$?A. 20412 Y 5 20412y^5 20412 Y 5 B. 35 A 4 B 3 35a^4b^3 35 A 4 B 3 C. 21 A 5 B 2 21a^5b^2 21 A 5 B 2 D. − 10206 Y 8 -10206y^8 − 10206 Y 8

Understanding the Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form $(a + b)^n$, where $a$ and $b$ are numbers or variables, and $n$ is a positive integer. The theorem states that the expansion of $(a + b)^n$ 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 Theorem to the Given Expression

In the given expression $(3y - 2)^7$, we have $a = 3y$ and $b = -2$. We want to find the value of the 3rd term of the expansion. Using the binomial theorem, we can write the expansion as:

$$(3y - 2)^7 = \binom{7}{0} (3y)^7 (-2)^0 + \binom{7}{1} (3y)^6 (-2)^1 + \binom{7}{2} (3y)^5 (-2)^2 + \ldots + \binom{7}{7} (3y)^0 (-2)^7$$

Finding the 3rd Term of the Expansion

To find the 3rd term of the expansion, we need to find the coefficient of the term that corresponds to the 3rd power of $b$. In this case, the 3rd term is:

$$\binom{7}{2} (3y)^5 (-2)^2$$

Calculating the Coefficient

To calculate the coefficient, we need to evaluate the binomial coefficient $\binom{7}{2}$ and the powers of $3y$ and $-2$.

$$\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2!5!} = \frac{7 \cdot 6}{2} = 21$$

$$(3y)^5 = 3^5 y^5 = 243y^5$$

$$(-2)^2 = 4$$

Combining the Terms

Now we can combine the terms to find the value of the 3rd term of the expansion:

$$\binom{7}{2} (3y)^5 (-2)^2 = 21 \cdot 243y^5 \cdot 4 = 20412y^5$$

Conclusion

Therefore, the value of the 3rd term of the expansion $(3y - 2)^7$ is $20412y^5$.

Discussion

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form $(a + b)^n$. In this problem, we used the binomial theorem to find the value of the 3rd term of the expansion $(3y - 2)^7$. We calculated the coefficient of the term that corresponds to the 3rd power of $b$ and combined the terms to find the final answer.

Common Mistakes

One common mistake that students make when applying the binomial theorem is to forget to calculate the binomial coefficient. Another mistake is to forget to evaluate the powers of the variables.

Tips for Solving Similar Problems

To solve similar problems, make sure to:

• Read the problem carefully and understand what is being asked.
• Identify the values of $a$, $b$, and $n$.
• Apply the binomial theorem to find the expansion.
• Calculate the binomial coefficient and the powers of the variables.
• Combine the terms to find the final answer.

Practice Problems

Try solving the following problems:

• Find the value of the 5th term of the expansion $(2x + 3)^8$.
• Find the value of the 2nd term of the expansion $(x - 2)^6$.
• Find the value of the 4th term of the expansion $(3x - 2)^9$.

Solutions

• The value of the 5th term of the expansion $(2x + 3)^8$ is $-3840x^3$.
• The value of the 2nd term of the expansion $(x - 2)^6$ is $-12x^5$.
• The value of the 4th term of the expansion $(3x - 2)^9$ is $-1944x^5$.

Conclusion

In conclusion, the binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form $(a + b)^n$. By applying the binomial theorem and calculating the binomial coefficient and the powers of the variables, we can find the value of any term in the expansion.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us 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 theorem?

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

1. Identify the values of $a$, $b$, and $n$.
2. Write the expansion using the formula: $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$
3. Calculate the binomial coefficient $\binom{n}{k}$ for each term.
4. Evaluate the powers of $a$ and $b$ for each term.
5. Combine the terms to find the final answer.

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that represents the number of ways to choose $k$ items from a set of $n$ items. It is defined as:

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

Q: How do I calculate the binomial coefficient?

A: To calculate the binomial coefficient, you need to:

1. Calculate the factorial of $n$ (denoted as $n!$).
2. Calculate the factorial of $k$ (denoted as $k!$).
3. Calculate the factorial of $(n-k)$ (denoted as $(n-k)!$).
4. Divide the result of step 1 by the product of the results of steps 2 and 3.

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

A: The binomial theorem is the formula that allows us to expand expressions of the form $(a + b)^n$. The binomial expansion is the actual expansion of the expression using the formula.

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

A: No, the binomial theorem is only applicable to expressions with non-negative exponents. If you have an expression with negative exponents, you need to use a different method to expand it.

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

A: No, the binomial theorem is only applicable to expressions with integer exponents. If you have an expression with fractional exponents, you need to use a different method to expand it.

Q: How do I find the value of a specific term in the expansion?

A: To find the value of a specific term in the expansion, you need to:

1. Identify the term number (starting from 0).
2. Calculate the binomial coefficient $\binom{n}{k}$ for that term.
3. Evaluate the powers of $a$ and $b$ for that term.
4. Combine the terms to find the final answer.

Q: Can I use the binomial theorem to expand expressions with complex numbers?

A: Yes, the binomial theorem can be used to expand expressions with complex numbers. However, you need to be careful when working with complex numbers and make sure to follow the correct rules for multiplying and dividing complex numbers.

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

A: Yes, the binomial theorem can be used to expand expressions with matrices. However, you need to be careful when working with matrices and make sure to follow the correct rules for matrix multiplication and division.

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 calculate the binomial coefficient.
• Forgetting to evaluate the powers of $a$ and $b$.
• Making mistakes when multiplying and dividing complex numbers or matrices.
• Not following the correct rules for matrix multiplication and division.

Q: How do I practice using the binomial theorem?

A: To practice using the binomial theorem, you can try:

• Expanding expressions with small values of $n$.
• Finding the value of specific terms in the expansion.
• Using the binomial theorem to solve problems in algebra and calculus.
• Working with complex numbers and matrices.

Q: What are some real-world applications of the binomial theorem?

A: The binomial theorem has many real-world applications, including:

• Statistics and probability: The binomial theorem is used to calculate probabilities and expected values in statistics and probability.
• Computer science: The binomial theorem is used in computer science to calculate the number of possible outcomes in a system.
• Engineering: The binomial theorem is used in engineering to calculate the stress and strain on materials.
• Finance: The binomial theorem is used in finance to calculate the value of options and other financial instruments.

Q: Can I use the binomial theorem to solve problems in other fields?

A: Yes, the binomial theorem can be used to solve problems in other fields, including:

• Physics: The binomial theorem is used in physics to calculate the probability of certain events.
• Chemistry: The binomial theorem is used in chemistry to calculate the concentration of solutions.
• Biology: The binomial theorem is used in biology to calculate the probability of certain genetic events.

Q: What are some tips for using the binomial theorem effectively?

A: Some tips for using the binomial theorem effectively include:

• Read the problem carefully and understand what is being asked.
• Identify the values of $a$, $b$, and $n$.
• Apply the binomial theorem to find the expansion.
• Calculate the binomial coefficient and the powers of $a$ and $b$.
• Combine the terms to find the final answer.
• Practice using the binomial theorem to build your skills and confidence.