Binomial Expansion

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What is Binomial Expansion?

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Binomial Expansion is a method used to expand expressions that are raised to a power, specifically in the form (a + b)^n, where 'a' and 'b' are any numbers, and 'n' is a non-negative integer.

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What is the Binomial Theorem?

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The Binomial Theorem provides a formula for the expansion of (a + b)^n, 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?

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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 constant term in a binomial expansion?

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To find the constant term in the expansion of (a + b)^n, set the variable part to zero and solve for the corresponding term.

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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) = C(n, k-1) * a^(n-(k-1)) * b^(k-1).

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

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To determine the coefficient of a specific term a^m * b^n in the expansion of (a + b)^n, use the formula C(n, n-m) * a^m * b^n.

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

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The fourth term is given by T(4) = C(5, 3) * x^(5-3) * y^3 = 10xy^3.

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How do you find the coefficient of x^3 in the expansion of (2x - 1/x^2)^12?

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Use the formula for the k-th term and set the exponent of x to 3, then solve for the coefficient.

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

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The coefficients in a binomial expansion represent the number of ways to choose terms from the binomial, and they follow the pattern of Pascal's Triangle.

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What is the relationship between the coefficients of (a + b)^n and Pascal's Triangle?

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The coefficients of (a + b)^n correspond to the n-th row of Pascal's Triangle, where each number is the sum of the two numbers directly above it.