Binomial Expansion Calculator
Expand binomial expressions using the Binomial Theorem with step-by-step solutions
Enter the terms and exponent to expand ( + )
Expanded Form of ( + )
terms using binomial coefficients
Term-by-Term Breakdown
Binomial Coefficients
These are the coefficients from row of Pascal's Triangle
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About Binomial Expansion Calculator
What is Binomial Expansion?
Binomial expansion is a method for expressing the power of a binomial (a two-term expression) as a sum of terms using the Binomial Theorem. It allows you to expand expressions like (a + b)^n into a polynomial without performing repeated multiplication.
The Binomial Theorem Formula
The Binomial Theorem states that for any non-negative integer n:
(a + b)^n = Σ C(n,k) × a^(n-k) × b^k for k = 0 to n
Where C(n,k) is the binomial coefficient, also written as "n choose k":
C(n,k) = n! / (k! × (n-k)!)
How to Use This Calculator
- Enter First Term (a): Input the coefficient or variable for the first term
- Enter Second Term (b): Input the coefficient or variable for the second term
- Enter Exponent (n): Input the power to raise the binomial to
- View Results: See the expanded form with all terms and coefficients
Pascal's Triangle and Binomial Coefficients
Pascal's Triangle provides a quick way to find binomial coefficients:
| Row | Coefficients |
|---|---|
| n=0 | 1 |
| n=1 | 1, 1 |
| n=2 | 1, 2, 1 |
| n=3 | 1, 3, 3, 1 |
| n=4 | 1, 4, 6, 4, 1 |
| n=5 | 1, 5, 10, 10, 5, 1 |
Common Binomial Expansions
- (a + b)² = a² + 2ab + b²
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a - b)² = a² - 2ab + b²
- (a - b)³ = a³ - 3a²b + 3ab² - b³
Frequently Asked Questions
What is the binomial coefficient?
The binomial coefficient C(n,k) tells you how many ways you can choose k items from n items. It's also the coefficient of the kth term in the binomial expansion.
Can I expand negative exponents?
The simple binomial theorem applies to non-negative integer exponents. For negative or fractional exponents, you need the generalized binomial theorem which produces infinite series.
What's the difference between (a+b)^n and (a-b)^n?
When expanding (a-b)^n, the signs alternate: positive for even powers of b, negative for odd powers of b.
How many terms are in a binomial expansion?
The expansion of (a+b)^n always has exactly (n+1) terms.
Note: This calculator handles non-negative integer exponents. For other cases, consult a more advanced tool.