Binomial Theorem Flashcard

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?

Back

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

Front

How do you find the 4th term in the expansion of (4x - 3)^5?

Back

The 4th term can be found using the formula T_k = (n choose k-1) * a^(n-k+1) * b^(k-1). For (4x - 3)^5, the 4th term is -4320x^2.

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, and it is used to find coefficients in binomial expansions.

Front

Expand (4 - 2z)^3 using the Binomial Theorem.

Back

(4 - 2z)^3 = 64 - 96z + 48z^2 - 8z^3.

Front

What is the third term in the expansion of (2x - 5)^5?

Back

The third term is 2000x^3.

Front

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

Back

The general term T_k in the expansion of (a + b)^n is given by T_k = (n choose k-1) * a^(n-k+1) * b^(k-1).

Front

What is the value of (n choose k)?

Back

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

Front

How do you calculate the coefficients in a binomial expansion?

Back

Coefficients in a binomial expansion can be calculated using the binomial coefficients from Pascal's Triangle or using the formula (n choose k).

Front

What is the significance of the Binomial Theorem in algebra?

Back

The Binomial Theorem simplifies the process of expanding binomials raised to a power, making it easier to compute and understand polynomial expressions.