Binomial Theorem

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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.

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

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(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.

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How do you find the third term in the expansion of (x + 4)^5 using the Binomial Theorem?

Back

The third term can be found using the formula: T(k+1) = C(n, k) * a^(n-k) * b^k. For (x + 4)^5, T(3) = C(5, 2) * x^(5-2) * 4^2 = 10 * x^3 * 16 = 160x^3.

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What is the general form of the expansion of (a + b)^n?

Back

The general form is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k = 0 to n.

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What is the coefficient of x^2 in the expansion of (x + 3)^3?

Back

The coefficient of x^2 in the expansion is 9, derived from the term C(3, 1) * x^2 * 3^1 = 3 * x^2 * 3 = 9x^2.

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How do you expand (2x + 5)^4 using the Binomial Theorem?

Back

Using the Binomial Theorem, (2x + 5)^4 = Σ (4 choose k) * (2x)^(4-k) * 5^k for k = 0 to 4, resulting in 16x^4 + 160x^3 + 600x^2 + 1000x + 625.

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What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of binomial coefficients, where each number is the sum of the two directly above it, used to find coefficients in binomial expansions.

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What is row 7 of Pascal's Triangle?

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Row 7 of Pascal's Triangle is 1, 7, 21, 35, 35, 21, 7, 1.

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How do you find the coefficient of x in the expansion of (3x - 4)^2?

Back

To find the coefficient of x, expand (3x - 4)^2 = 9x^2 - 24x + 16. The coefficient of x is -24.

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What is the significance of the coefficients in the Binomial Theorem?

Back

The coefficients in the Binomial Theorem correspond to the values in Pascal's Triangle and represent the number of ways to choose terms in the expansion.