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), 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. It is calculated as n! / (k!(n-k)!).

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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: C(n, k) * a^(n-k) * b^k, where a and b are the terms in the binomial, n is the exponent, and k is the power of x.

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Expand (x - 3)^10 using the Binomial Theorem.

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(x - 3)^10 = Σ (10 choose k) * x^(10-k) * (-3)^k for k = 0 to 10.

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What is the 9th term in the expansion of (2x - 5)^{11}?

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The 9th term corresponds to k = 8: C(11, 8) * (2x)^(11-8) * (-5)^8 = 515625000x^3.

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How do you calculate the coefficient of x^2 in (2x - 3)^9?

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Use k = 2: C(9, 2) * (2x)^(9-2) * (-3)^2 = -314928.

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

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

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

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The signs alternate based on the power of b. If n is even, the last term is positive; if n is odd, the last term is negative.

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Find the coefficient of x^9 in (3x - 1)^{12}.

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The coefficient of x^9 corresponds to k = 3: C(12, 3) * (3x)^9 * (-1)^3 = -4330260.

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What is the formula for the coefficient of x^k in (a + b)^n?

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C(n, k) * a^(n-k) * b^k.