Find The Coefficient Of $x^3$ In The Expansion Of The Binomial Expression $(1+3x)^{-2}$.A. -648 B. 108 C. 648 D. -108

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Introduction

The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$, where $a$ and $b$ are constants and $n$ is a positive integer. However, the theorem can also be extended to handle negative exponents and non-integer exponents. In this article, we will explore how to find the coefficient of $x^3$ in the expansion of the binomial expression $(1+3x)^{-2}$.

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:

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

Extending the Binomial Theorem to Negative Exponents

To extend the binomial theorem to handle negative exponents, we can use the following formula:

$$(a+b)^{-n} = \frac{1}{(a+b)^n} = \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} a^{-n-k} b^k$$

This formula allows us to expand expressions of the form $(a+b)^{-n}$, where $n$ is a positive integer.

Applying the Binomial Theorem to $(1+3x)^{-2}$

To find the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$, we can use the extended binomial theorem formula. We have:

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

Finding the Coefficient of $x^3$

To find the coefficient of $x^3$, we need to find the term in the expansion that contains $x^3$. We can do this by setting $k=3$ in the formula above:

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

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

$$= -\binom{4}{3} 1^{-5} (3x)^3 + \cdots$$

$$= -\frac{4!}{3!(4-3)!} 1^{-5} (3x)^3 + \cdots$$

$$= -\frac{4!}{3!1!} 1^{-5} (3x)^3 + \cdots$$

$$= -\frac{4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1} 1^{-5} (3x)^3 + \cdots$$

$$= -4 \cdot 1^{-5} (3x)^3 + \cdots$$

$$= -4 \cdot (3x)^3 + \cdots$$

$$= -4 \cdot 27x^3 + \cdots$$

$$= -108x^3 + \cdots$$

Conclusion

In this article, we used the extended binomial theorem formula to find the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$. We showed that the coefficient of $x^3$ is $-108$. This result can be verified by using other methods, such as using the binomial theorem to expand the expression and then collecting the terms that contain $x^3$.

Final Answer

The final answer is $\boxed{-108}$.

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Introduction

In our previous article, we explored how to find the coefficient of $x^3$ in the expansion of the binomial expression $(1+3x)^{-2}$. We used the extended binomial theorem formula to derive the coefficient of $x^3$ as $-108$. In this article, we will answer some common questions related to this topic.

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$, where $a$ and $b$ are constants and $n$ is a positive integer. It states that:

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

Q: How do I apply the binomial theorem to negative exponents?

A: To apply the binomial theorem to negative exponents, we can use the following formula:

$$(a+b)^{-n} = \frac{1}{(a+b)^n} = \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} a^{-n-k} b^k$$

Q: What is the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$?

A: The coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$ is $-108$. This can be derived using the extended binomial theorem formula.

Q: How do I find the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$?

A: To find the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$, you can use the following steps:

1. Write down the expression $(1+3x)^{-2}$.
2. Use the extended binomial theorem formula to expand the expression.
3. Collect the terms that contain $x^3$.
4. Simplify the expression to find the coefficient of $x^3$.

Q: What is the significance of the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$?

A: The coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$ is significant because it represents the rate of change of the function $(1+3x)^{-2}$ with respect to $x$. In other words, it represents the slope of the function at a given point.

Q: Can I use the binomial theorem to expand expressions with non-integer exponents?

A: Yes, you can use the binomial theorem to expand expressions with non-integer exponents. However, the formula for the binomial coefficient will be different.

Q: How do I apply the binomial theorem to expressions with non-integer exponents?

A: To apply the binomial theorem to expressions with non-integer exponents, you can use the following formula:

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

where $n$ is a non-integer.

Conclusion

In this article, we answered some common questions related to finding the coefficient of $x^3$ in the expansion of $(1+3x)^{-2}$. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{-108}$.