1. What is the Binomial Theorem?

The Binomial Theorem describes the algebraic expansion of powers of a binomial (an expression of the form a + b). According to the theorem, it is possible to expand the polynomial (a + b)ⁿ into a sum involving terms of the form C(n,k)a^n-kb^k.

2. How Does the Calculator Work?

The calculator uses the Binomial Theorem formula:

Where:

• \( a \) and \( b \) — Terms of the binomial
• \( n \) — The exponent (must be a non-negative integer)
• \( \binom{n}{k} \) — Binomial coefficient (n choose k)

Explanation: The theorem expands the expression into a sum of terms where the exponents of a decrease from n to 0 while exponents of b increase from 0 to n.

3. Applications of Binomial Expansion

Details: Binomial expansion is used in probability, statistics, algebra, calculus, and many areas of mathematics and science. It's fundamental for polynomial approximations and series expansions.

4. Using the Calculator

Tips: Enter values for a and b (can be numbers or variables), and a non-negative integer for n. The calculator will show the expanded form with all terms.

5. Frequently Asked Questions (FAQ)

Q1: What is the maximum exponent (n) allowed?
A: The calculator supports exponents up to 100 for practical purposes.

Q2: Can I use variables instead of numbers?
A: Yes, the calculator works with both numerical and symbolic expressions.

Q3: How are the binomial coefficients calculated?
A: Coefficients are calculated using the formula C(n,k) = n!/(k!(n-k)!).

Q4: What happens when n=0?
A: Any non-zero number to the power of 0 is 1, so (a+b)⁰ = 1.

Q5: Can this be used for negative exponents?
A: No, this calculator only handles non-negative integer exponents.