Question

Binomial expansion

Answer 1

[AI-generated answer — Bot GPT 3.5]

The binomial expansion is a mathematical formula that allows us to expand a binomial expression raised to a positive integer power.

The general form of the binomial expansion is:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

Where C(n, k) represents the binomial coefficient, which can be calculated using the formula:

C(n, k) = n! / (k! * (n-k)!)

In this formula, "a" and "b" are the binomial terms, "n" is the power of the binomial, and "C(n, k)" represents the binomial coefficient. The formula allows us to expand the binomial expression by considering all possible combinations of the two terms raised to different powers.

For example, if we have the expression (x + y)^3, we can use the binomial expansion to expand it as:

(x + y)^3 = C(3, 0) * x^3 * y^0 + C(3, 1) * x^2 * y^1 + C(3, 2) * x^1 * y^2 + C(3, 3) * x^0 * y^3

Simplifying this expression, we get:

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

The binomial expansion is a very useful tool in algebra and calculus, as it allows us to find the terms of a binomial expression raised to any positive integer power without having to manually calculate each term.