☔ 17.9 Binomial Theorem

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What is the Binomial Theorem?

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The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k for k = 0 to n.

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

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(n choose k) or C(n, k) represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated as C(n, k) = n! / (k!(n-k)!) where '!' denotes factorial.

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How do you find the coefficient of a specific term in a binomial expansion?

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To find the coefficient of x^k in the expansion of (a + b)^n, use the formula: Coefficient = (n choose k) * a^(n-k) * b^k, where a and b are the terms in the binomial.

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Expand (x + 2)^4 using the Binomial Theorem.

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Using the Binomial Theorem, (x + 2)^4 = Σ (4 choose k) * x^(4-k) * 2^k for k = 0 to 4. The expansion is: x^4 + 8x^3 + 24x^2 + 32x + 16.

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What is the 5th term in the expansion of (3x + 4)^6?

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The 5th term corresponds to k = 4. Using the formula: Coefficient = (6 choose 4) * (3x)^(6-4) * (4)^4 = 15 * 9x^2 * 256 = 34560x^2.

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Find the coefficient of x^2 in the expansion of (2x + 3)^5.

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The coefficient of x^2 is given by: (5 choose 2) * (2x)^2 * 3^(5-2) = 10 * 4x^2 * 27 = 1080.

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What is the significance of the expansion of (a - b)^n?

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The expansion of (a - b)^n follows the same structure as (a + b)^n, but the signs alternate. It is given by: (a - b)^n = Σ (n choose k) * a^(n-k) * (-b)^k.

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Use the Binomial Theorem to expand (2x - 3)^3.

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(2x - 3)^3 = (2x)^3 - 3*(2x)^2*(3) + 3*(2x)*(3^2) - (3^3) = 8x^3 - 36x^2 + 54x - 27.

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What is the formula for the k-th term in the expansion of (a + b)^n?

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

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Find the coefficient of x^4 in the expansion of (x + 5)^6.

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The coefficient of x^4 is given by: (6 choose 4) * (5)^2 = 15 * 25 = 375.