Definition

Noun: A fundamental theorem in algebra that provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. It describes the algebraic expansion of powers of a binomial (a two-term expression).

Usage

The binomial theorem is used to expand binomial expressions raised to a power without performing lengthy multiplications. It is a key concept in algebra, probability, and calculus.

Examples

• According to the binomial theorem, (a + b)^2 expands to a^2 + 2ab + b^2.
• To find the coefficient of the x^3 term in the expansion of (2x - 1)^5, we applied the binomial theorem.
• The binomial theorem states that (x + y)^n = Σ [n! / (k!(n-k)!)] * x^(n-k) * y^k, where k goes from 0 to n.

Advanced Usage

• Generalized Binomial Theorem: This extends the theorem to allow for exponents that are not non-negative integers (like fractions or negative numbers), resulting in an infinite series expansion.
  • Example: The generalized binomial theorem can be used to find approximations for roots, such as √(1+x).
• In Probability (Binomial Distribution): The coefficients from the binomial expansion (the binomial coefficients) directly give the probabilities in a binomial distribution.
  • Example: The probability of getting exactly k heads in n coin tosses is given by a term from the binomial theorem expansion.

Variants and Related Words

• Binomial (n/adj): A polynomial with exactly two terms (e.g., x+1). Also used to describe the two-term nature of the expression or related distributions.
• Binomial Coefficient (n): The numerical factor in the terms of the binomial expansion, denoted as "n choose k" or C(n, k) = n! / (k!(n-k)!).
• Binomial Expansion (n): The result of applying the binomial theorem; the expanded form of (x + y)^n.

Synonyms

• Binomial Expansion Formula

Related Phrases/Concepts

• Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it. The rows of Pascal's Triangle give the binomial coefficients for the expansion of (x + y)^n.
  • Example: The coefficients for (a+b)^4 can be read directly from the 5th row of Pascal's Triangle: 1, 4, 6, 4, 1.

Noun

1. a theorem giving the expansion of a binomial raised to a given power