A) Find The First Three Terms, In Ascending Powers Of X X X , Of The Binomial Expansion Of ( 2 + P X ) 7 (2 + P X)^7 ( 2 + P X ) 7 , Where P P P Is A Constant.b) The First Three Terms Are 128 128 128 , 2240 X 2240x 2240 X , And Q X 2 Q X^2 Q X 2 . Find

Introduction

The binomial expansion is a mathematical concept used to expand expressions of the form $(a + b)^n$, where $a$ and $b$ are constants and $n$ is a positive integer. In this article, we will explore the binomial expansion of the expression $(2 + p x)^7$, where $p$ is a constant. We will find the first three terms of the expansion in ascending powers of $x$.

The Binomial Theorem

The binomial theorem is a formula that describes the expansion of expressions of the form $(a + b)^n$. The theorem states that:

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

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

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

Finding the First Three Terms

To find the first three terms of the binomial expansion of $(2 + p x)^7$, we will use the binomial theorem. We will start by expanding the expression using the theorem:

$$(2 + p x)^7 = \binom{7}{0} 2^7 (p x)^0 + \binom{7}{1} 2^6 (p x)^1 + \binom{7}{2} 2^5 (p x)^2 + \ldots + \binom{7}{7} 2^0 (p x)^7$$

Simplifying the expression, we get:

$$(2 + p x)^7 = 128 + 2240 p x + 3360 p^2 x^2 + \ldots + p^7 x^7$$

Therefore, the first three terms of the binomial expansion of $(2 + p x)^7$ are:

• $128$
• $2240 p x$
• $3360 p^2 x^2$

Solving for $q$

We are given that the first three terms of the binomial expansion are $128$, $2240x$, and $qx^2$. We can equate the coefficients of the terms to find the value of $q$. Equating the coefficients of the $x^2$ term, we get:

$$3360 p^2 = q$$

Therefore, the value of $q$ is:

$$q = 3360 p^2$$

Conclusion

In this article, we have found the first three terms of the binomial expansion of $(2 + p x)^7$, where $p$ is a constant. We have also solved for the value of $q$ by equating the coefficients of the terms. The binomial expansion is a powerful tool for expanding expressions of the form $(a + b)^n$, and it has many applications in mathematics and other fields.

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica. Retrieved from https://www.britannica.com/topic/binomial-theorem
• [2] Binomial Expansion. (n.d.). In Math Open Reference. Retrieved from https://www.mathopenref.com/binomial_expansion.html

Further Reading

• [1] Binomial Theorem. (n.d.). In Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/BinomialTheorem.html
• [2] Binomial Expansion. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5b7d7/binomial-expansion

Mathematical Notation

• $\binom{n}{k}$: binomial coefficient
• $n!$: factorial of $n$
• $a^n b^m$: exponentiation of $a$ and $b$
• $(a + b)^n$: binomial expansion of $(a + b)^n$
  Binomial Expansion: Q&A =========================

Introduction

In our previous article, we explored the binomial expansion of the expression $(2 + p x)^7$, where $p$ is a constant. We found the first three terms of the expansion in ascending powers of $x$ and solved for the value of $q$. In this article, we will answer some frequently asked questions about the binomial expansion.

Q: What is the binomial theorem?

A: The binomial theorem is a formula that describes the expansion of expressions of the form $(a + b)^n$, where $a$ and $b$ are constants and $n$ is a positive integer.

Q: How do I use the binomial theorem to expand an expression?

A: To use the binomial theorem to expand an expression, you need to follow these steps:

1. Write the expression in the form $(a + b)^n$.
2. Use the formula for the binomial coefficient to find the coefficients of the terms.
3. Simplify the expression by combining like terms.

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the binomial theorem. It is defined as:

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

Q: How do I find the value of $q$ in the binomial expansion?

A: To find the value of $q$ in the binomial expansion, you need to equate the coefficients of the terms. For example, if the first three terms of the expansion are $128$, $2240x$, and $qx^2$, you can equate the coefficients of the $x^2$ term to find the value of $q$.

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

A: The binomial expansion has many applications in mathematics and other fields. Some common applications include:

• Expanding expressions of the form $(a + b)^n$
• Finding the coefficients of the terms in a binomial expansion
• Solving equations involving binomial expansions
• Finding the value of $q$ in a binomial expansion

Q: How do I use the binomial expansion to solve equations?

A: To use the binomial expansion to solve equations, you need to follow these steps:

1. Write the equation in the form $(a + b)^n = c$.
2. Use the binomial theorem to expand the left-hand side of the equation.
3. Simplify the equation by combining like terms.
4. Solve for the unknown variable.

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

A: Some common mistakes to avoid when using the binomial expansion include:

• Not using the correct formula for the binomial coefficient
• Not simplifying the expression by combining like terms
• Not equating the coefficients of the terms to find the value of $q$
• Not using the binomial expansion to solve equations

Conclusion

In this article, we have answered some frequently asked questions about the binomial expansion. We have covered topics such as the binomial theorem, the binomial coefficient, and the value of $q$ in a binomial expansion. We have also discussed some common applications of the binomial expansion and some common mistakes to avoid when using it.

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica. Retrieved from https://www.britannica.com/topic/binomial-theorem
• [2] Binomial Expansion. (n.d.). In Math Open Reference. Retrieved from https://www.mathopenref.com/binomial_expansion.html

Further Reading

• [1] Binomial Theorem. (n.d.). In Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/BinomialTheorem.html
• [2] Binomial Expansion. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5b7d7/binomial-expansion

Mathematical Notation

• $\binom{n}{k}$: binomial coefficient
• $n!$: factorial of $n$
• $a^n b^m$: exponentiation of $a$ and $b$
• $(a + b)^n$: binomial expansion of $(a + b)^n$