4.5 Warm Up: Pascal's Triangle & Binomial Theorem

Front

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.

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 for k = 0 to n.

Front

Expand (b - 2)^3 using the Binomial Theorem.

Back

b^3 - 6b^2 + 12b - 8.

Front

What is the coefficient of x^3 in the expansion of (x - 3)^10?

Back

-262440.

Front

How do you find the coefficients in a binomial expansion?

Back

Use the binomial coefficients from Pascal's Triangle or calculate using the formula (n choose k).

Front

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

Back

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

Front

Which row of Pascal's Triangle corresponds to (x + y)^3?

Back

1 3 3 1.

Front

How do you use Pascal's Triangle to expand (2x + 5)^4?

Back

Use the coefficients from the 4th row of Pascal's Triangle: 1, 4, 6, 4, 1.

Front

What is the general form of a binomial expansion?

Back

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k.

Front

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

Back

The coefficients represent the number of ways to choose k elements from n, which corresponds to the number of combinations.