The Binomial Theorem provides a formula for expanding expressions that are raised to a power. Specifically, it describes the expansion of a binomial, which is an expression of the form $(a + b)^n$, where $a$ and $b$ are any numbers or variables, and $n$ is a positive integer.

Binomial Theorem Formula:

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

• $\binom{n}{k}$ is a binomial coefficient, also known as "n choose k," which is calculated as: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• $n!$ (n factorial) is the product of all positive integers from 1 to $n$.
• The summation $\sum$ runs from $k = 0$ to $n$, and each term represents a part of the expansion.

Explanation:

• The binomial expansion creates a sum of terms, where each term has the form $\binom{n}{k} a^{n-k} b^k$. The exponents of $a$ decrease from $n$ to 0, while the exponents of $b$ increase from 0 to $n$.
• The coefficients $\binom{n}{k}$ determine how much each term contributes to the expansion.

Example: Expand $(x + 2)^3$

To expand $(x + 2)^3$, we can apply the binomial theorem.

• $a = x$, $b = 2$, and $n = 3$.

The expansion formula is: $(x + 2)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} 2^k$ Now, let's compute the terms:

1. For $k = 0$: $\binom{3}{0} x^{3-0} 2^0 = 1 \cdot x^3 \cdot 1 = x^3$

2. For $k = 1$: $\binom{3}{1} x^{3-1} 2^1 = 3 \cdot x^2 \cdot 2 = 6x^2$

3. For $k = 2$: $\binom{3}{2} x^{3-2} 2^2 = 3 \cdot x^1 \cdot 4 = 12x$

4. For $k = 3$: $\binom{3}{3} x^{3-3} 2^3 = 1 \cdot x^0 \cdot 8 = 8$

Now, putting all the terms together: $(x + 2)^3 = x^3 + 6x^2 + 12x + 8$

Summary:

The expansion of $(x + 2)^3$ is: $x^3 + 6x^2 + 12x + 8$

Key Points to Remember:

1. The binomial theorem applies to any binomial expression raised to a power $n$.
2. The expansion involves terms of the form $\binom{n}{k} a^{n-k} b^k$.
3. Each term in the expansion involves both a power of $a$ and a power of $b$, and the binomial coefficients $\binom{n}{k}$ are crucial for determining the coefficients of the terms.

---

Would you like me to go into more details on any part of the process, or do you have any specific questions?

Here are some related questions:

1. How do you calculate binomial coefficients for larger values of $n$?
2. What happens when $a = b$ in the binomial expansion?
3. How do you expand expressions like $(x - 3)^4$ using the binomial theorem?
4. Can the binomial theorem be used for negative or fractional exponents?
5. What is the pattern of binomial coefficients, and how can you find them without expanding the whole expression?

Tip: Practice expanding binomials with different values of $n$ to become more comfortable with the binomial theorem and recognizing the patterns in the expansions.