Binomial Theorem

Front

What is the Binomial Theorem?

Back

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.

Front

What does (2x + 5)^4 expand to using the Binomial Theorem?

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(2x + 5)^4 = 16x^4 + 160x^3 + 600x^2 + 1000x + 625.

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What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of the binomial coefficients. Each number is the sum of the two directly above it.

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What is row 7 of Pascal's Triangle?

Back

1, 7, 21, 35, 35, 21, 7, 1.

Front

What is the formula for the binomial coefficient (n choose k)?

Back

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

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How do you calculate (a + b)^3 using the Binomial Theorem?

Back

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

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What is the significance of the coefficients in the expansion of (a + b)^n?

Back

The coefficients correspond to the values in row n of Pascal's Triangle.

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

Back

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

Front

Expand (x + 2)^5 using the Binomial Theorem.

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(x + 2)^5 = x^5 + 10x^4 * 2 + 40x^3 * 4 + 80x^2 * 8 + 32.

Front

What is the value of (3 + 4)^2 using the Binomial Theorem?

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(3 + 4)^2 = 3^2 + 2*3*4 + 4^2 = 9 + 24 + 16 = 49.