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 (m - 3)^5 represent in the context of the Binomial Theorem?

Back

It represents a binomial expression raised to the 5th power, which can be expanded using the Binomial Theorem.

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

Back

Use the formula for the k-th term: T(k+1) = (n choose k) * a^(n-k) * b^k. For (m - 3)^5, the 6th term is T(6) = (5 choose 5) * m^(5-5) * (-3)^5 = -243.

Front

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

Back

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

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What is the value of (n choose k)?

Back

It is calculated as n! / (k! * (n-k)!), representing the number of ways to choose k elements from a set of n elements.

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Expand (d - 5)^5 using the Binomial Theorem.

Back

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

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What is the Binomial expansion of (x + 1)^5?

Back

x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

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How do you determine the 2nd term of the expansion of (y + 4)^4?

Back

Use T(2) = (4 choose 1) * y^(4-1) * 4^1 = 16y^3.

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What is the 5th term of the expansion of (n + 4)^4?

Back

The 5th term is T(5) = (4 choose 4) * n^(4-4) * 4^4 = 256.

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

Back

The coefficients correspond to the binomial coefficients (n choose k) and represent the number of ways to choose k elements from n.