1. What Is Binomial Expansion?

The binomial theorem describes the algebraic expansion of powers of a binomial (an expression with two terms). It provides a way to expand expressions of the form (a + b)ⁿ without directly multiplying the binomial by itself n times.

2. How Does the Calculator Work?

The calculator uses the binomial theorem formula:

Where:

• \( a \) — First term in the binomial
• \( b \) — Second term in the binomial
• \( n \) — Power of the expansion
• \( k \) — Term number (0 to n)
• \( \binom{n}{k} \) — Binomial coefficient (n choose k)

Explanation: The (k+1)th term in the expansion is given by the binomial coefficient multiplied by a to the power of (n-k) and b to the power of k.

3. Importance of Binomial Expansion

Details: Binomial expansion is fundamental in algebra, probability, and calculus. It's used in probability theory (binomial distribution), series approximations, and solving polynomial equations.

4. Using the Calculator

Tips: Enter the two terms of your binomial (can be variables or numbers), the power (n), and the term number (k) you want to find (starting from 0). The calculator will compute the (k+1)th term in the expansion.

5. Frequently Asked Questions (FAQ)

Q1: What if k is greater than n?
A: The binomial coefficient is zero for k > n, so the term would be zero.

Q2: Can I use negative exponents?
A: This calculator is designed for non-negative integer exponents only.

Q3: How is the binomial coefficient calculated?
A: It's calculated as n!/(k!(n-k)!), where ! denotes factorial.

Q4: Can I use this for fractional or negative terms?
A: Yes, the binomial theorem works for any terms a and b, as long as n is a non-negative integer.

Q5: What's the difference between term number and term index?
A: Term numbers typically start at 1 (first term), while term indices start at 0. This calculator uses term index (k).