What is the binomial theorem?

The binomial theorem is a formula for expanding expressions of the form (a + b)^n.

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n, where n is a positive integer. The formula for the binomial theorem 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)a^0 b^n

where C(n,k) is the binomial coefficient, given by:

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

The binomial coefficient represents the number of ways to choose k objects from a set of n objects. For example, C(4,2) = 6, because there are 6 ways to choose 2 objects from a set of 4 objects.

To use the binomial theorem, we simply substitute the values of a, b, and n into the formula and simplify. For example, to expand (x + y)^3, we have:

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

The binomial theorem can also be used to find specific terms in the expansion of (a + b)^n. For example, to find the coefficient of x^2 y^3 in the expansion of (x + 2y)^5, we would use the formula:

C(5,2)(x^2)(2y)^3 = 10x^2 y^3

Overall, the binomial theorem is a useful tool for simplifying and expanding algebraic expressions, and is an important concept for A-Level Maths students to understand.