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Binomial theorem Calculator

Expand (a+b)^n for a positive integer n.

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Term index k

Formula first

Overview

The binomial theorem provides an efficient algebraic method for expanding expressions of the form (a + b) raised to any non-negative integer power n. It expresses the expansion as a sum of n + 1 terms, where each term's coefficient is a binomial coefficient calculated using combinations.

Symbols

Variables

a = First Term, b = Second Term, n = Power n, k = Term index k, T = Term Value

First Term
Second Term
Power n
Term index k
Term Value

Apply it well

When To Use

When to use: Use this theorem when you need to expand a binomial expression without performing repeated multiplication. It is also used to find a specific term in an expansion or to calculate probabilities in binomial distributions where only certain outcomes are relevant.

Why it matters: This theorem is a fundamental bridge between algebra and combinatorics, providing the basis for the Binomial Distribution in statistics. It is used in physics for power series approximations and in computer science for analyzing the complexity of recursive algorithms.

Avoid these traps

Common Mistakes

  • Forgetting coefficients.
  • Wrong powers summing to n.

One free problem

Practice Problem

In the expansion of (a + 2)³, the second term (where k = 1) evaluates to 24. Solve for the value of the variable a.

Second Term2
Power n3
Term index k1
Term Value24

Solve for:

Hint: Use the term formula: C(n, k) × aⁿ⁻ᵏ × bᵏ.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Binomial theorem
  2. Stewart, James. Calculus. Cengage Learning.
  3. Britannica: Binomial theorem
  4. AQA A-Level Mathematics — Pure (Binomial Expansion)