Determine The Coefficient Of $x^4$ In The Binomial Expansion $\left(4x - \frac{1}{2}\right)^7$.

Introduction to Binomial Expansion

Binomial expansion is a mathematical concept that deals with the expansion of expressions in the form of $(a + b)^n$. It is a fundamental concept in algebra and is used to find the value of expressions that involve powers of a binomial. 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, which is defined as:

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

Understanding the Problem

In this problem, we are given the expression $\left(4x - \frac{1}{2}\right)^7$ and we need to find the coefficient of $x^4$ in its binomial expansion. To do this, we need to use the binomial expansion formula and find the term that involves $x^4$.

Finding the Term Involving $x^4$

To find the term involving $x^4$, we need to find the term in the binomial expansion that has $x^4$ as its power. Since the power of $x$ in the given expression is $4$, we need to find the term that has $x^4$ as its power.

Using the binomial expansion formula, we can write the expression as:

$$\left(4x - \frac{1}{2}\right)^7 = \binom{7}{0} (4x)^7 \left(-\frac{1}{2}\right)^0 + \binom{7}{1} (4x)^6 \left(-\frac{1}{2}\right)^1 + \binom{7}{2} (4x)^5 \left(-\frac{1}{2}\right)^2 + \cdots + \binom{7}{7} (4x)^0 \left(-\frac{1}{2}\right)^7$$

Calculating the Coefficient of $x^4$

To find the coefficient of $x^4$, we need to find the term that involves $x^4$. This term is given by:

$$\binom{7}{3} (4x)^4 \left(-\frac{1}{2}\right)^3$$

Using the binomial coefficient formula, we can calculate the value of $\binom{7}{3}$ as:

$$\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$$

Calculating the Coefficient of $x^4$

Now that we have the value of $\binom{7}{3}$, we can calculate the coefficient of $x^4$ as:

$$\binom{7}{3} (4x)^4 \left(-\frac{1}{2}\right)^3 = 35 \times (4x)^4 \times \left(-\frac{1}{2}\right)^3$$

$$= 35 \times 256x^4 \times \left(-\frac{1}{8}\right)$$

$$= 35 \times 256 \times \left(-\frac{1}{8}\right) x^4$$

$$= -\frac{8960}{8} x^4$$

$$= -1120 x^4$$

Conclusion

In this article, we have determined the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$. We have used the binomial expansion formula and calculated the value of the binomial coefficient to find the coefficient of $x^4$. The final answer is $-1120 x^4$.

Frequently Asked Questions

• What is the binomial expansion formula? The binomial expansion formula is given by: $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$
• What is the binomial coefficient? The binomial coefficient is defined as: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• How do I find the coefficient of $x^4$ in the binomial expansion? To find the coefficient of $x^4$, you need to find the term in the binomial expansion that has $x^4$ as its power. You can use the binomial expansion formula and calculate the value of the binomial coefficient to find the coefficient of $x^4$.

References

• Binomial expansion formula: $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$
• Binomial coefficient formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
  

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Introduction

In our previous article, we determined the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$. In this article, we will answer some frequently asked questions related to binomial expansion and the coefficient of $x^4$.

Q1: What is the binomial expansion formula?

A1: The binomial expansion formula is given by:

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

Q2: What is the binomial coefficient?

A2: The binomial coefficient is defined as:

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

Q3: How do I find the coefficient of $x^4$ in the binomial expansion?

A3: To find the coefficient of $x^4$, you need to find the term in the binomial expansion that has $x^4$ as its power. You can use the binomial expansion formula and calculate the value of the binomial coefficient to find the coefficient of $x^4$.

Q4: What is the formula for calculating the binomial coefficient?

A4: The formula for calculating the binomial coefficient is:

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

Q5: How do I calculate the value of the binomial coefficient?

A5: To calculate the value of the binomial coefficient, you need to use the formula:

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

For example, to calculate the value of $\binom{7}{3}$, you can use the formula:

$$\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$$

Q6: How do I find the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$?

A6: To find the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$, you need to use the binomial expansion formula and calculate the value of the binomial coefficient. The term involving $x^4$ is given by:

$$\binom{7}{3} (4x)^4 \left(-\frac{1}{2}\right)^3$$

Using the binomial coefficient formula, we can calculate the value of $\binom{7}{3}$ as:

$$\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$$

Q7: What is the final answer for the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$?

A7: The final answer for the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$ is:

$$-1120 x^4$$

Conclusion

In this article, we have answered some frequently asked questions related to binomial expansion and the coefficient of $x^4$. We have provided the binomial expansion formula, the binomial coefficient formula, and the formula for calculating the value of the binomial coefficient. We have also provided examples of how to calculate the value of the binomial coefficient and how to find the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$.

Frequently Asked Questions

• What is the binomial expansion formula?
• What is the binomial coefficient?
• How do I find the coefficient of $x^4$ in the binomial expansion?
• What is the formula for calculating the binomial coefficient?
• How do I calculate the value of the binomial coefficient?
• How do I find the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$?
• What is the final answer for the coefficient of $x^4$ in the binomial expansion $\left(4x - \frac{1}{2}\right)^7$?

References

• Binomial expansion formula: $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$
• Binomial coefficient formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$