Find The Coefficient Of $x^4$ In The Expansion Of $\left(1+\frac{1}{2} X\right)^8$.

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Introduction

The binomial theorem is a powerful tool in mathematics 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. In this article, we will use the binomial theorem to find the coefficient of $x^4$ in the expansion of the expression $\left(1+\frac{1}{2} x\right)^8$.

The Binomial Theorem

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

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

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

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

Applying the Binomial Theorem

To find the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$, we can use the binomial theorem with $a = 1$, $b = \frac{1}{2} x$, and $n = 8$. Plugging these values into the formula, we get:

$$\left(1+\frac{1}{2} x\right)^8 = \sum_{k=0}^{8} \binom{8}{k} 1^{8-k} \left(\frac{1}{2} x\right)^k$$

Finding the Term with $x^4$

To find the term with $x^4$, we need to find the value of $k$ that makes the power of $x$ equal to 4. Since the power of $x$ is given by $k$, we can set $k = 4$ to get the term with $x^4$:

$$\binom{8}{4} 1^{8-4} \left(\frac{1}{2} x\right)^4 = \binom{8}{4} \left(\frac{1}{2} x\right)^4$$

Calculating the Binomial Coefficient

To calculate the binomial coefficient $\binom{8}{4}$, we can use the formula:

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

Plugging in the values $n = 8$ and $k = 4$, we get:

$$\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}$$

Simplifying the Factorial Expressions

To simplify the factorial expressions, we can use the fact that $n! = n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 2 \cdot 1$. Plugging in the values, we get:

$$8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$$

$$4! = 4 \cdot 3 \cdot 2 \cdot 1$$

$$4! = 4 \cdot 3 \cdot 2 \cdot 1$$

Calculating the Binomial Coefficient

Now that we have simplified the factorial expressions, we can calculate the binomial coefficient:

$$\binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(4 \cdot 3 \cdot 2 \cdot 1)(4 \cdot 3 \cdot 2 \cdot 1)}$$

Simplifying the Expression

To simplify the expression, we can cancel out the common factors in the numerator and denominator:

$$\binom{8}{4} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1}$$

Calculating the Final Value

Now that we have simplified the expression, we can calculate the final value of the binomial coefficient:

$$\binom{8}{4} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1} = 70$$

Finding the Coefficient of $x^4$

Now that we have calculated the binomial coefficient, we can find the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$:

$$\binom{8}{4} \left(\frac{1}{2} x\right)^4 = 70 \left(\frac{1}{2} x\right)^4 = 70 \cdot \frac{1}{16} x^4 = \frac{70}{16} x^4$$

Conclusion

In this article, we used the binomial theorem to find the coefficient of $x^4$ in the expansion of the expression $\left(1+\frac{1}{2} x\right)^8$. We calculated the binomial coefficient $\binom{8}{4}$ and used it to find the coefficient of $x^4$. The final answer is $\boxed{\frac{70}{16}}$.

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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 to find the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$?

A: To apply the binomial theorem, you need to plug in the values $a = 1$, $b = \frac{1}{2} x$, and $n = 8$ into the formula. This will give you the expansion of the expression, and you can then find the term with $x^4$.

Q: What is the binomial coefficient, and how do I calculate it?

A: The binomial coefficient is a number that appears in the binomial theorem formula. It is calculated using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the power of the binomial and $k$ is the term number.

Q: How do I simplify the factorial expressions in the binomial coefficient formula?

A: To simplify the factorial expressions, you can use the fact that $n! = n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 2 \cdot 1$. You can then cancel out the common factors in the numerator and denominator to simplify the expression.

Q: What is the final value of the binomial coefficient $\binom{8}{4}$?

A: The final value of the binomial coefficient $\binom{8}{4}$ is 70.

Q: How do I find the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$?

A: To find the coefficient of $x^4$, you need to multiply the binomial coefficient $\binom{8}{4}$ by the term $\left(\frac{1}{2} x\right)^4$. This will give you the coefficient of $x^4$ in the expansion of the expression.

Q: What is the final answer for the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$?

A: The final answer for the coefficient of $x^4$ in the expansion of $\left(1+\frac{1}{2} x\right)^8$ is $\boxed{\frac{70}{16}}$.

Q: Can I use the binomial theorem to find the coefficient of $x^4$ in other binomial expressions?

A: Yes, you can use the binomial theorem to find the coefficient of $x^4$ in other binomial expressions. You just need to plug in the values of $a$, $b$, and $n$ into the formula and follow the same steps as before.

Q: What are some common applications of the binomial theorem?

A: The binomial theorem has many common applications in mathematics, including algebra, calculus, and probability theory. It is used to expand expressions, find coefficients, and solve equations.

Q: Can I use the binomial theorem to find the coefficient of $x^4$ in a binomial expression with negative powers?

A: Yes, you can use the binomial theorem to find the coefficient of $x^4$ in a binomial expression with negative powers. However, you need to be careful when simplifying the expression and calculating the binomial coefficient.

Q: What are some tips for using the binomial theorem to find the coefficient of $x^4$ in a binomial expression?

A: Some tips for using the binomial theorem to find the coefficient of $x^4$ in a binomial expression include:

• Make sure to plug in the correct values of $a$, $b$, and $n$ into the formula.
• Simplify the factorial expressions carefully to avoid errors.
• Use the binomial coefficient formula to calculate the binomial coefficient.
• Multiply the binomial coefficient by the term to find the coefficient of $x^4$.
• Check your work carefully to ensure that you have the correct answer.