Lesson 17.1 DOL Binomial Expansions

Front

What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for the expansion of powers of a binomial (a + b)^n, expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.

Front

How many terms are in the expansion of (a + b)^n?

Back

There are n + 1 terms in the expansion of (a + b)^n.

Front

What is a binomial coefficient?

Back

A binomial coefficient, denoted as C(n, k) or nCk, represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.

Front

What is the formula for calculating binomial coefficients?

Back

C(n, k) = n! / (k! * (n - k)!), where n! is the factorial of n.

Front

What is the expansion of (1 + x)^3?

Back

1 + 3x + 3x^2 + x^3.

Front

What is the expansion of (a + b)^2?

Back

a^2 + 2ab + b^2.

Front

What is the third term in the expansion of (x + y)^5?

Back

10x^3y^2.

Front

What is the expansion of (2x - 3)^4?

Back

16x^4 - 96x^3 + 216x^2 - 162x + 81.

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) = C(n, k-1) * a^(n-(k-1)) * b^(k-1).

Front

What is the significance of the signs in the expansion of (1 - x)^n?

Back

The signs alternate based on the power of -1, leading to terms that are positive and negative depending on the term's position.