Binomial Theorem
Expands any power of a binomial into a sum of terms.
Formula
Variables
Example
(x+1)³ = x³ + 3x² + 3x + 1
How it works
The binomial theorem tells you how to expand a power of a two-term expression like (a+b)^n without multiplying it out repeatedly. Each term in the expansion comes from choosing either a or b from each of the n factors, and the coefficient \(\binom{n}{k}\) counts how many ways to choose exactly k b's and the remaining a's. That counting idea is why the pattern of coefficients matches Pascal's triangle.
When to use it
Use it whenever you need to expand a binomial raised to a whole-number power, especially for algebraic simplification, finding specific terms, or comparing coefficients. It is also useful in probability and combinatorics, where the same coefficients count combinations.
Worked example, step by step
Expand (2x - 3)^3 Step 1: Identify a = 2x, b = -3, and n = 3 Step 2: Apply the binomial theorem: (2x - 3)^3 = \binom{3}{0}(2x)^3(-3)^0 + \binom{3}{1}(2x)^2(-3)^1 + \binom{3}{2}(2x)^1(-3)^2 + \binom{3}{3}(2x)^0(-3)^3 Step 3: Evaluate each term: = 1·8x^3·1 + 3·4x^2·(-3) + 3·2x·9 + 1·1·(-27) Step 4: Simplify: = 8x^3 - 36x^2 + 54x - 27 Final answer: (2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
Common mistakes to avoid
- Forgetting that the powers of a and b must add up to n in every term
- Missing the alternating signs when b is negative
- Using the wrong binomial coefficient or writing the terms in the wrong order
Did You Know?
The coefficients form Pascal's Triangle, known in Persia, China, and India centuries before Pascal was born.
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