Binomial Theorem

Front

What is the Binomial Theorem?

Back

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

Front

What does (a + b)^4 expand to?

Back

(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.

Front

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

Back

The k-th term is given by T_k = (n choose (k-1)) * a^(n-(k-1)) * b^(k-1).

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How do you find the 4th term of (a + 3)^4?

Back

Use the formula: T_4 = (4 choose 3) * a^(4-3) * 3^3 = 1 * a^1 * 27 = 27a.

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What is the expansion of (b - 2)^3?

Back

(b - 2)^3 = b^3 - 6b^2 + 12b - 8.

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What is the expansion of (x - 7)^2?

Back

(x - 7)^2 = x^2 - 14x + 49.

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What is the expansion of (d - 5)^5?

Back

(d - 5)^5 = d^5 - 25d^4 + 250d^3 - 1250d^2 + 3125d - 3125.

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How do you find the 6th term of (m - 3)^5?

Back

Use the formula: T_6 = (5 choose 5) * m^(5-5) * (-3)^5 = 1 * 1 * (-243) = -243.

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

Back

The coefficients in the expansion correspond to the binomial coefficients, which can be found using Pascal's Triangle or the formula (n choose k).

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What is Pascal's Triangle?

Back

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