Please solve the problem by binomial theorem rule.I can not express it.

Let's dive into the Binomial Theorem and see how we can use it to solve problems involving binomial expansions. The Binomial Theorem provides a way to expand expressions of the form (a + b)^n, where 'a' and 'b' are any numbers, and 'n' is a non-negative integer. The theorem states that:

Understanding the Binomial Theorem

The expansion of (a + b)^n can be expressed as:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

Here, the summation (Σ) runs from k = 0 to n, and (n choose k) is the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Breaking It Down

To clarify, let’s break down the components:

• n! (n factorial) is the product of all positive integers up to n.
• k! is the factorial of k.
• (n-k)! is the factorial of the difference between n and k.

Each term in the expansion corresponds to a specific value of k, which tells us how many times to multiply 'b' and how many times to multiply 'a'.

Example Problem

Let’s say we want to expand (x + 2)^3 using the Binomial Theorem. Here, a = x, b = 2, and n = 3.

Step-by-Step Expansion

Using the theorem, we can write:

(x + 2)^3 = Σ (3 choose k) * x^(3-k) * 2^k

Now, we will calculate each term for k = 0, 1, 2, and 3:

• For k = 0: (3 choose 0) * x^(3-0) * 2^0 = 1 * x^3 * 1 = x^3
• For k = 1: (3 choose 1) * x^(3-1) * 2^1 = 3 * x^2 * 2 = 6x^2
• For k = 2: (3 choose 2) * x^(3-2) * 2^2 = 3 * x^1 * 4 = 12x
• For k = 3: (3 choose 3) * x^(3-3) * 2^3 = 1 * 1 * 8 = 8

Combining the Terms

Now, we combine all these terms together:

(x + 2)^3 = x^3 + 6x^2 + 12x + 8

Final Thoughts

The Binomial Theorem is a powerful tool for expanding binomials quickly and efficiently. By understanding how to apply the theorem, you can tackle a variety of problems involving polynomial expansions. If you have any specific examples or further questions, feel free to ask!