Binomial Expansion

Front

What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for the expansion of powers of a binomial, expressed as (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.

Front

What does (n choose k) represent in the Binomial Theorem?

Back

(n choose k), denoted as C(n, k) or nCk, represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.

Front

How do you find the k-th term in the expansion of (a + b)^n?

Back

The k-th term in the expansion of (a + b)^n is given by T(k) = C(n, k-1) * a^(n-(k-1)) * b^(k-1), where k starts from 1.

Front

What is the fourth term in the expansion of (5 + 3y)^5?

Back

The fourth term is 6750y^3.

Front

What is the expansion of (1 + 4x)^3?

Back

The expansion is 1 + 12x + 48x^2 + 64x^3.

Front

What is the expansion of (1 - x)^5?

Back

The expansion is 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5.

Front

What is the Binomial expansion of (x + 1)^5?

Back

The expansion is x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

Front

How do you find the second term in the expansion of (2b + 1)^3?

Back

The second term is found using T(2) = C(3, 1) * (2b)^(3-1) * (1)^1 = 12b^2.

Front

What is the general form of a binomial expansion?

Back

The general form of a binomial expansion is (a + b)^n = a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + b^n.

Front

What is the significance of the coefficients in a binomial expansion?

Back

The coefficients in a binomial expansion correspond to the binomial coefficients, which represent the number of ways to choose terms from the expansion.