Binomial Expansion

Front

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.

Front

What does (x + 1)^5 expand to using the Binomial Theorem?

Back

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

Front

How do you find 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).

Front

What is Pascal's Triangle and how is it used in binomial expansion?

Back

Pascal's Triangle is a triangular array of binomial coefficients. Each row corresponds to the coefficients in the expansion of (a + b)^n.

Front

What is the significance of the coefficients in the expansion of (x + y)^n?

Back

The coefficients represent the number of ways to choose terms from the binomial expansion and are derived from the corresponding row in Pascal's Triangle.

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What is the formula for the binomial coefficient, often denoted as (n choose k)?

Back

The binomial coefficient is calculated as (n choose k) = n! / (k!(n-k)!), where n! is the factorial of n.

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What is the fourth term in the expansion of (5 + 3y)^5?

Back

The fourth term is 6750y^3.

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How do you determine the third term in the expansion of (x + 4)^5?

Back

The third term can be found using the formula T(3) = (5 choose 2) * x^(5-2) * 4^2 = 160x^3.

Front

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

Back

The general form is Σ (n choose k) * a^(n-k) * b^k for k = 0 to n.

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What is the relationship between the terms in the expansion of (x + y)^n?

Back

The terms are symmetric, meaning T(k) = T(n-k) for any k.